### Packages used in this chapter

The packages used in this chapter include:

• psych

• DescTools

• Rmisc

• FSA

• plyr

• boot

The following commands will install these packages if they are not already installed:

if(!require(psych)){install.packages("psych")}

if(!require(DescTools)){install.packages("DescTools")}

if(!require(Rmisc)){install.packages("Rmisc")}

if(!require(FSA)){install.packages("FSA")}

if(!require(plyr)){install.packages("plyr")}

if(!require(boot)){install.packages("boot")}

### Descriptive statistics

Descriptive statistics are used to summarize data in a way that provides insight into the information contained in the data. This might include examining the mean or median of numeric data or the frequency of observations for nominal data. Plots can be created that show the data and indicating summary statistics.

Choosing which summary statistics are appropriate depend on the type of variable being examined. Different statistics should be used for interval/ratio, ordinal, and nominal data.

In describing or examining data, you will typically be concerned with measures of location, variation, and shape.

*Location* is also called *central tendency*.
It is a measure of the values of the data. For example, are the
values close to 10 or 100 or 1000? Measures of location include
mean and median, as well as somewhat more exotic statistics like M-estimators
or Winsorized means.

*Variation* is also called *dispersion*.
It is a measure of how far the data points lie from one another.
Common statistics include standard deviation and coefficient of variation.
For data that aren’t normally-distributed, percentiles or the interquartile
range might be used.

*Shape* refers to the distribution of values.
The best tools to evaluate the shape of data are histograms and related
plots. Statistics include skewness and kurtosis, though they are
less useful than visual inspection. We can describe data shape
as normally-distributed, log-normal, uniform, skewed, bi-modal, and
others.

### Descriptive statistics for interval/ratio data

For this example, imagine that Ren and Stimpy have each held eight workshops educating the public about water conservation at home. They are interested in how many people showed up to the workshops.

Because the data are housed in a data frame, we
can use the convention *Data$Attendees* to access the variable
*Attendees* within the data frame *Data*.

Input = ("

Instructor Location Attendees

Ren North 7

Ren North
22

Ren North 6

Ren North
15

Ren South
12

Ren South
13

Ren South
14

Ren South
16

Stimpy North
18

Stimpy North
17

Stimpy North
15

Stimpy North 9

Stimpy South 15

Stimpy South 11

Stimpy South 19

Stimpy South 23

")

Data = read.table(textConnection(Input),header=TRUE)

Data ###
Will output data frame called Data

str(Data) ###
Will be discussed later,

###
but shows the structure of the data frame

summary(Data)
### Will be discussed later,

### but summarizes variables in the data frame

##### Functions *sum* and *length*

The sum of a variable can be found with the *
sum* function, and the number of observations can be found with the
*length* function.

sum(Data$Attendees)

232

length(Data$Attendees)

16

#### Statistics of location for interval/ratio data

##### Mean

The mean is the arithmetic average, and is a common
statistic used with interval/ratio data. It is simply the sum
of the values divided by the number of values. The *mean*
function in R will return the mean.

sum(Data$Attendees) / length(Data$Attendees)

14.5

mean(Data$Attendees)

14.5

Caution should be used when reporting mean values with skewed data, as the mean may not be representative of the center of the data. For example, imagine a town with 10 families, nine of whom have an income of less than $50, 000 per year, but with one family with an income of $2,000,000 per year. The mean income for families in the town would be $233,000, but this may not be a reasonable way to summarize the income of the town.

Income = c(49000, 44000, 25000, 18000, 32000, 47000,
37000, 45000, 36000, 2000000)

mean(Income)

233300

##### Median

The median is defined as the value below which are 50% of the observations. To find this value manually, you would order the observations, and separate the lowest 50% from the highest 50%. For data sets with an odd number of observations, the median is the middle value. For data sets with an even number of observations, the median falls half-way between the two middle values.

The median is a robust statistic in that it is
not affected by adding extreme values. For example, if we changed
Stimpy’s last *Attendees* value from 23 to 1000, it would not affect
the median.

median(Data$Attendees)

15

### Note that in this case the mean and median are close in value

### to one another. The mean and median will be more
different

### the more the data are skewed.

The median is appropriate for either skewed or unskewed data. The median income for the town discussed above is $40,500. Half the families in the town have an income above this amount, and half have an income below this amount.

Income = c(49000, 44000, 25000, 18000, 32000, 47000, 37000,
45000, 36000, 2000000)

median(Income)

40500

Note that medians are sometimes reported as the “average person” or “typical family”. Saying, “The average American family earned $54,000 last year” means that the median income for families was $54,000. The “average family” is that one with the median income.

##### Mode

The mode is a summary statistic that is used rarely
in practice, but is normally included in any discussion of mean and
medians. When there are discreet values for a variable, the mode
is simply the value which occurs most frequently. For example,
in the Statistics Learning Center video in the *Required Readings*
below, Dr. Nic gives an example of counting the number of pairs of shoes
each student owns. The most common answer was 10, and therefore
10 is the mode for that data set.

For our Ren and Stimpy example, the value 15 occurs three times and so is the mode.

The *Mode* function can be found in the package
*DescTools*.

library(DescTools)

Mode(Data$Attendees)

15

#### Statistics of variation for interval/ratio data

##### Standard deviation

The standard deviation is a measure of variation which is commonly used with interval/ratio data. It’s a measurement of how close the observations in the data set are to the mean.

There’s a handy rule of thumb that—for normally distributed data—68% of data points fall within the mean ± 1 standard deviation, 95% of data points fall within the mean ± 2 standard deviations, and 99.7% of data points fall within the mean ± 3 standard deviations.

Because the mean is often represented with the
letter *mu*, and the standard deviation is represented with the
letter *sigma*, saying someone is “a few *sigmas* away from
*mu*” indicates they are rather a rare character. (I originally
heard this joke on an episode of *Car Talk*, for which I cannot
find a reference or transcript.)

sd(Data$Attendees)

4.830459

Standard deviation may not be appropriate for skewed data.

##### Standard error of the mean

Standard error of the mean is a measure that estimates how close a calculated mean is likely to be to the true mean of that population. It is commonly used in tables or plots where multiple means are presented together. For example, we might want to present the mean attendees for Ren with the standard error for that mean and the mean attendees for Stimpy with the standard error that mean.

The standard error is the standard deviation of
a data set divided by the square root of the number of observations.
It can also be found in the output for the *describe* function
in the *psych* package, labelled *se*.

sd(Data$Attendees) /

sqrt(length(Data$Attendees))

1.207615

library(psych)

describe(Data$Attendees)

vars n mean sd median trimmed
mad min max range skew kurtosis se

1
1 16 14.5 4.83 15 14.5 4.45
6 23 17 -0.04 -0.88 1.21

### se indicates
the standard error of the mean

Standard error of the mean may not be appropriate for skewed data.

##### Five-number summary, quartiles, percentiles

The median is the same as the 50^{th} percentile,
because 50% of values fall below this value. Other percentiles
for a data set can be identified to provide more information.
Typically, the 0^{th}, 25^{th}, 50^{th}, 75^{th},
and 100^{th} percentiles are reported. This is sometimes
called the five-number summary.

These values can also be called the minimum, 1^{st}
quartile, 2^{nd} quartile, 3^{rd} quartile, and maximum.

The five-number summary is a useful measure of
variation for skewed interval/ratio data or for ordinal data.
25% of values fall below the 1^{st} quartile and 25% of values
fall above the 3^{rd} quartile. This leaves the middle
50% of values between the 1^{st} and 3^{rd} quartiles,
giving a sense of the range of the middle half of the data. This
range is called the *interquartile range* (IQR).

Percentiles and quartiles are relatively robust, as they aren’t affected much by a few extreme values. They are appropriate for both skewed and unskewed data.

summary(Data$Attendees)

Min. 1st Qu. Median
Mean 3rd Qu. Max.

6.00
11.75 15.00 14.50 17.25
23.00

### The five-number summary and the mean

##### Optional technical note on calculating percentiles

It may have struck you as odd that the 3^{rd}
quartile for *Attendees *was reported as 17.25. After all,
if you were to order the values of *Attendees*, the 75^{th}
percentile would fall between 17 and 18. But why does R go with
17.25 and not 17.5?

sort(Data$Attendees)

6 7 9 11 12 13 14 15 15 15 16 17
18 19 22 23

The answer is that there are several different
methods to calculate percentiles, and they may give slightly different
answers. For details on the calculations, see *?quantiles*.

For *Attendees*, the default type 7 calculation
yields a 75^{th} percentile value of 17.25, whereas the type
2 calculation simply splits the difference between 17 and 18 and yields
17.5. The type 1 calculation doesn’t average the two values, and
so just returns 17.

quantile(Data$Attendees, 0.75, type=7)

75%

17.25

quantile(Data$Attendees, 0.75, type=2)

75%

17.5

quantile(Data$Attendees, 0.75, type=1)

75%

17

Percentiles other than the 25^{th},
50^{th}, and 75^{th} can be calculated with the quantiles
function. For example, to calculate the 95^{th} percentile:

quantile(Data$Attendees, .95)

95%

22.25

##### Confidence intervals

Confidence intervals are covered in the next chapter.

#### Statistics for grouped interval/ratio data

In many cases, we will want to examine summary statistics for a variable within groups. For example, we may want to examine statistics for the workshops lead by Ren and those lead by Stimpy.

*Summarize* in *FSA*

The *Summarize* function in the *FSA*
package returns the number of observations, mean, standard deviation,
minimum, 1^{st} quartile, median, 3^{rd} quartile, and
maximum for grouped data.

Note the use of formula notation: *Attendees*
is the dependent variable (the variable you want to get the statistics
for); and *Instructor* is the independent variable (the grouping
variable). *Summarize* allows you to summarize over the combination
of multiple independent variables by listing them to the right of the
~ separated by a plus sign (+).

library(FSA)

Summarize(Attendees ~ Instructor,

data=Data)

Instructor n nvalid mean
sd min Q1 median Q3 max percZero

1 Ren 8
8 13.125 5.083236 6 10.75 13.5 15.25 22
0

2 Stimpy 8
8 15.875 4.454131 9 14.00 16.0 18.25 23
0

Summarize(Attendees ~ Instructor + Location,

data=Data)

Instructor Location n nvalid mean
sd min Q1 median Q3 max percZero

1 Ren North
4 4 12.50 7.505554 6
6.75 11.0 16.75 22
0

2 Stimpy North 4
4 14.75 4.031129 9 13.50 16.0 17.25 18
0

3 Ren
South 4 4 13.75 1.707825 12 12.75
13.5 14.50 16 0

4
Stimpy South 4 4 17.00
5.163978 11 14.00 17.0 20.00 23
0

*summarySE* in *Rmisc*

The *summarySE* function in the *Rmisc*
package outputs the number of observations, mean, standard deviation,
standard error of the mean, and confidence interval for grouped data.
The *summarySE *function allows you to summarize over the combination
of multiple independent variables by listing them as a vector, e.g.
*c("Instructor", "Student")*.

library(Rmisc)

summarySE(data=Data,

"Attendees",

groupvars="Instructor",

conf.interval = 0.95)

Instructor N Attendees
sd se
ci

1 Ren 8
13.125 5.083236 1.797195 4.249691

2 Stimpy
8 15.875 4.454131 1.574773 3.723747

summarySE(data=Data,

"Attendees",

groupvars = c("Instructor", "Location"),

conf.interval = 0.95)

Instructor Location N Attendees
sd se ci

1 Ren North
4 12.50 7.505553 3.7527767 11.943011

2
Ren South 4 13.75 1.707825
0.8539126 2.717531

3 Stimpy
North 4 14.75 4.031129 2.0155644 6.414426

4 Stimpy South 4
17.00 5.163978 2.5819889 8.217041

*describeBy* in *psych*

The *describeBy *function in the *psych*
package returns the number of observations, mean, median, trimmed means,
minimum, maximum, range, skew, kurtosis, and standard error of the mean
for grouped data. *describeBy* allows you to summarize over
the combination of multiple independent variables by combining terms
with a colon (:).

library(psych)

describeBy(Data$Attendees,

group
= Data$Instructor,

digits= 4)

group: Ren

vars n
mean sd median trimmed mad min max range skew kurtosis
se

1 1 8 13.12 5.08 13.5
13.12 2.97 6 22 16 0.13
-1.08 1.8

-------------------------------------------------------------------------

group: Stimpy

vars n mean sd median trimmed
mad min max range skew kurtosis se

1
1 8 15.88 4.45 16 15.88 3.71
9 23 14 -0.06 -1.26 1.57

describeBy(Data$Attendees,

group = Data$Instructor : Data$Location,

digits= 4)

group: Ren:North

vars n mean sd median trimmed mad min max range skew
kurtosis se

1 1 4 12.5 7.51
11 12.5 6.67 6 22
16 0.26 -2.14 3.75

-------------------------------------------------------------------------

group: Ren:South

vars n mean sd median
trimmed mad min max range skew kurtosis se

1
1 4 13.75 1.71 13.5 13.75 1.48 12
16 4 0.28 -1.96 0.85

-------------------------------------------------------------------------

group: Stimpy:North

vars n mean sd median
trimmed mad min max range skew kurtosis se

1 1 4 14.75 4.03 16
14.75 2.22 9 18 9 -0.55
-1.84 2.02

-------------------------------------------------------------------------

group: Stimpy:South

vars n mean sd median
trimmed mad min max range skew kurtosis se

1
1 4 17 5.16 17
17 5.93 11 23 12 0
-2.08 2.58

#### Summaries for data frames

Often we will want to summarize variables in a whole data frame, either to get some summary statistics for individual variables, or to check that variables have the values we expected in inputting the data, to be sure there wasn’t some error.

##### The *str* function

The *str* function in the native *utils
*package will list variables for a data frame, with their types and
levels. *Data* is the name of the data frame, created above.

str(Data)

'data.frame': 16 obs. of
3 variables:

$ Instructor: Factor w/ 2 levels "Ren","Stimpy":
1 1 1 1 1 1 1 1 2 2 ...

$ Location : Factor w/ 2 levels
"North","South": 1 1 1 1 2 2 2 2 1 1 ...

$ Attendees : int
7 22 6 15 12 13 14 16 18 17 ...

### Instructor is a factor (nominal) variable
with two levels.

### Location is a factor (nominal)
variable with two levels.

### Attendees is an integer
variable.

##### The *summary* function

The *summary* function in the native *base*
package will summarize all variables in a data frame, listing the frequencies
for levels of nominal variables; and for interval/ratio data, the minimum,
1^{st} quartile, median, mean, 3^{rd} quartile, and
maximum.

summary(Data)

Instructor Location Attendees

Ren :8 North:8 Min.
: 6.00

Stimpy:8 South:8
1st Qu.:11.75

Median :15.00

Mean :14.50

3rd Qu.:17.25

Max. :23.00

##### The *headTail* function in *psych*

The *headTail *function in the *psych*
package reports the first and last observations for a data frame.

library(psych)

headTail(Data)

Instructor Location Attendees

1 Ren
North 7

2
Ren North
22

3 Ren
North 6

4
Ren North
15

... <NA> <NA>
...

13 Stimpy South
15

14 Stimpy South
11

15 Stimpy South
19

16 Stimpy South
23

##### The *describe* function in *psych*

The *describe* function in the *psych*
package reports number of observations, mean, standard deviation, trimmed
means, minimum, maximum, range, skew, kurtosis, and standard error for
variables in a data frame.

Note that factor variables are labeled with an asterisk (*), and the levels of the factors are coded as 1, 2, 3, etc.

library(psych)

describe(Data)

vars n mean sd median trimmed mad min max range
skew kurtosis se

Instructor* 1 16
1.5 0.52 1.5 1.5 0.74
1 2 1 0.00
-2.12 0.13

Location* 2 16 1.5
0.52 1.5 1.5 0.74
1 2 1 0.00
-2.12 0.13

Attendees 3 16 14.5 4.83
15.0 14.5 4.45 6 23
17 -0.04 -0.88 1.21

#### Dealing with missing values

Sometimes a data set will have missing values. This can occur for a variety of reasons, such as a respondent not answering a specific question, or a researcher being unable to make a measurement due to temporarily malfunctioning equipment.

In R, a missing value is indicated with *NA*.

By default, different functions in R will handle missing values in different ways. But most have options to change how they treat missing data.

In general, you should scan your data for missing data, and think carefully about the best way to handle observations with missing values.

Input = ("

Instructor Location Attendees

Ren North
7

Ren North
22

Ren North
6

Ren North
15

Ren South
12

Ren South
13

Ren South
**NA**

Ren
South 16

Stimpy
North 18

Stimpy
North 17

Stimpy
North **NA**

Stimpy North
9

Stimpy South
15

Stimpy South
11

Stimpy South
19

Stimpy South
23

")

Data2 = read.table(textConnection(Input),header=TRUE)

### Note:
This data frame will be called Data2 to distinguish it

### from Data
above.

Data2

##### The *na.rm* option for missing values with a simple function

Many common functions in R have a *na.rm*
option. If this option is set to *FALSE*, the function will
return an *NA* result if there are any NA’s in the data values
passed to the function. If set to *TRUE*, observations with
NA values will be discarded, and the function will continue to calculate
its output.

Note that the *na.rm* option operates only
on the data values actually passed to the function. In the following
example with *median*, only *Attendees* is passed to the function;
if there were NA’s in other variables, this would not affect the function.

Not all functions have the same default for the
*na.rm* option. To determine the default, use e.g. *?median*,
*?mean*, *?sd*.

median(Data2$Attendees,

na.rm = FALSE)

NA

### na.rm=FALSE. Since there is an NA in the data, report NA.

median(Data2$Attendees,

na.rm = TRUE)

15

### na.rm=TRUE. Drop observations with NA and then calculate the
median.

##### Missing values with *Summarize* in *FSA*

*Summarize* in *FSA* will indicate invalid
values, including NA’s, with the count of valid observations outputted
as the variable *nvalid*.

library(FSA)

Summarize(Attendees ~ Instructor,

data=Data2)

Instructor n nvalid mean
sd min Q1 median Q3 max percZero

1
Ren 8 7 13 5.477226
6 9.5 13 15.5 22
0

2 Stimpy 8
7 16 4.795832 9 13.0
17 18.5 23 0

### This function removes
missing values, but indicates the number of

###
missing values by not including them in the count for nvalid.

##### Missing values indicated with summary function

The *summary* function counts NA’s for numeric
variables.

summary(Data2)

Instructor Location Attendees

Ren :8 North:8 Min.
: 6.00

Stimpy:8 South:8
1st Qu.:11.25

Median :15.00

Mean :14.50

3rd Qu.:17.75

Max. :23.00

NA's :2

### Indicates two NA’s in Attendees.

##### Missing values in the describe function in psych

The *describe* function is the *psych*
package removes NA’s by default.

library(psych)

describe(Data2$Attendees)

vars n mean sd median trimmed
mad min max range skew kurtosis se

1
1 14 14.5 5.19 15 14.5 5.19
6 23 17 -0.04 -1.17 1.39

### Note that two
NA’s were removed by default, reporting an n of 14.

##### Missing values in the *summarySE* function in *Rmisc*

By default, the *summarySE* function does
not remove NA’s, but can be made to do so with the *na.rm=TRUE*
option.

library(Rmisc)

summarySE(data=Data2,

"Attendees")

.id N Attendees sd se ci

1 <NA>
16 NA NA NA NA

### Note an N of 16 is reported, and
statistics are reported as NA.

library(Rmisc)

summarySE(data=Data2,

"Attendees",

na.rm=TRUE)

.id N Attendees
sd se
ci

1 <NA> 14 14.5 5.185038 1.38576
2.993752

### Note
an N of 14 is reported, and statistics are calculated with

### NA’s removed.

##### Advanced techniques

###### Discarding subjects

Certain psychology studies use scales that are calculated from responses of several questions. Because answers to all questions are needed to reliably calculate the scale, any observation (subject, person) with missing answers will simply be discarded. Some types of analyses in other fields also follow this approach.

Subjects can be deleted with the *subset *
function. The following code creates a new data frame, *Data3*,
with all observations with *NA* in the variable *Attendees*
removed from Data2.

Data3 = subset(Data2,

!is.na(Attendees))

Data3

###### Imputation of values

Missing values can be assigned a likely value through
the process of imputation. An algorithm is used that determines
a value based on the values of other variables for that observation
relative to values for other variables. The *mice* package
in R can perform imputation.

##### Optional code: removing missing values in vectors

If functions don’t have a *na.rm* option,
it is possible to remove NA observations manually. Here we’ll
create a vector called *valid* that just contains those values
of *Attendees* that are not NA’s.

The *!* operator is a logical *not*.
The brackets serve as a list of observations to include for the preceding
variable. So, the code essentially says, “Define a vector *valid*
as the values of *Attendees* where the values of *Attendees*
are *not* *NA*.”

valid = Data2$Attendees[!is.na(Data2$Attendees)]

summary(valid)

Min. 1st Qu. Median Mean 3rd
Qu. Max.

6.00 11.25
15.00 14.50 17.75 23.00

#### Statistics of shape for interval/ratio data

The most common statistics for shape are skewness and kurtosis.

Understanding skewness and kurtosis are important as they are ways in which a distribution of data varies from a normal distribution. This will be important in assessing the assumptions of certain statistical tests.

However, I rarely see skewness and kurtosis values reported. Instead, normality is usually assessed visually with plot, or using certain statistical tests. One problem with using skewness and kurtosis values is that there is not agreement in what values constitute meaningful deviations from the normal curve.

##### Skewness

Skewness indicates the degree of asymmetry in a data set. If there are relatively more values that are far greater than the mean, the distribution is positively skewed or right skewed, with a tail stretching to the right. Negative or left skew is the opposite.

A symmetric distribution has a skewness of 0. The skewness value for a positively skewed distribution is positive, and a negative value for a negatively skewed distribution. Sometimes a skew with an absolute value greater than 1 or 1.5 or 2 is considered highly skewed. There is not agreement on this interpretation.

There are different methods to calculate skew and
kurtosis. The *describe* function in the *psych* package
has three options for how to calculate them.

library(psych)

describe(Data$Attendees,

type=3)
### Type of calculation for skewness
and kurtosis

vars n mean sd median trimmed
mad min max range skew kurtosis se

1
1 16 14.5 4.83 15 14.5 4.45
6 23 17 -0.04 -0.88 1.21

### Skewness and kurtosis
among other statistics

The normal curve is symmetrical
around its center. The positively-skewed distribution has a longer,
thicker tail to the right. The negatively-skewed distribution
has a longer, thicker tail to the left.

For more information on the normal curve, see the video on “Normal Distribution” from Statistics Learning Center in the “Optional Readings” section. Additional thoughts on the normal distribution and real-world data distributions are in the article by Dr. Nic in the “Optional Readings” section.

##### Kurtosis

Kurtosis measures the degree to which the distribution has either fewer and less extreme outliers, or more and more extreme outliers. In general, the higher the kurtosis, the sharper the peak and the longer the tails. This is called leptokurtic, and is indicated by positive kurtosis values. The opposite—platykurtosis—has negative kurtosis values.

Confusingly, the kurtosis for a normal distribution is sometimes defined as 0, and sometimes defined as 3. The former is often called “excess kurtosis”.

Sometimes an excess kurtosis with an absolute value greater than 2 or 3 is considered a high deviation from being mesokurtic. There is not agreement on this interpretation.

### Descriptive statistics for ordinal data

Descriptive statistics for ordinal data are more limited than those for interval/ratio data. You’ll remember that for ordinal data, the levels can be ordered, but we can’t say that the intervals between the levels are equal. For example, we can’t say that an Associate’s degree and Master’s degree somehow average out to a Bachelor’s degree. This concept is discussed in more detail in the chapters on Likert data.

Because of this fact, several common descriptive statistics are usually inappropriate for use with ordinal data. These include mean, standard deviation, and standard error of the mean.

Ordinal data can be described by either: 1) treating the data as numeric and using appropriate statistics such as median and quartiles; or, 2) treating the data as nominal, and looking at counts of the data for each level.

A more-complete discussion of descriptive statistics
for ordinal data can be found in the *Descriptive Statistics for Likert
Data* chapter.

#### Example of descriptive statistics for ordinal data

For this example, imagine that Arthur and Baxter have each held a workshop educating the public about preventing water pollution at home. They are interested in the education level of people who attended the workshops. We can consider education level an ordinal variable. They also collected information on the sex and county of participants.

For this data, we have manually coded *Education*
with numbers in a separate variable *Ed.code*. These numbers
list the education in order: High school < Associate’s <
Bachelor’s < Master’s < Ph.D.

Of course we could have had R do this coding for us, but the code is slightly messy, so I did the coding manually, and listed the optional R code below.

Ideally we would want to treat *Education* as an
*ordered factor* variable in R. But unfortunately most common
functions in R won’t handle ordered factors well. Later in this
book, ordered factor data will be handled directly with cumulative
link models (CLM), permutation tests, and tests for ordered tables.

Optional R code is shown below for converting a factor to an
ordered factor.

Input = ("

Date
Instructor Student Sex
County Education Ed.code

'2015-11-01' 'Arthur Read' a female
'Elwood' BA
3

'2015-11-01' 'Arthur Read' b
female 'Bear Lake' PHD
5

'2015-11-01' 'Arthur Read' c
male 'Elwood' BA
3

'2015-11-01' 'Arthur Read' d
female 'Elwood' MA
4

'2015-11-01' 'Arthur Read' e
male 'Elwood' HS
1

'2015-11-01' 'Arthur Read' f
female 'Bear Lake' MA
4

'2015-11-01' 'Arthur Read' g
male 'Elwood' HS
1

'2015-11-01' 'Arthur Read' h
female 'Elwood' BA
3

'2015-11-01' 'Arthur Read' i
female 'Elwood' BA
3

'2015-11-01' 'Arthur Read' j
female 'Elwood' BA
3

'2015-12-01' 'Buster Baxter' k
male 'Elwood' MA 4

'2015-12-01' 'Buster Baxter' l
male 'Bear Lake' MA
4

'2015-12-01' 'Buster Baxter' m
female 'Elwood' AA
2

'2015-12-01' 'Buster Baxter' n
male 'Elwood' AA
2

'2015-12-01' 'Buster Baxter' o
other 'Elwood' BA
3

'2015-12-01' 'Buster Baxter' p
female 'Elwood' BA
3

'2015-12-01' 'Buster Baxter' q
female 'Bear Lake' PHD
5

")

Data = read.table(textConnection(Input),header=TRUE)

### Check the data
frame

Data

str(Data)

'data.frame': 17 obs. of
7 variables:

$ Date : Factor w/
2 levels "2015-11-01","2015-12-01": 1 1 1 1 1 1 1 1 1 1

$
Instructor: Factor w/ 2 levels "Arthur Read",..: 1 1 1 1 1 1 1 1 1 1
...

$ Student : Factor w/ 17 levels "a","b","c","d",..:
1 2 3 4 5 6 7 8 9 10 ...

$ Sex
: Factor w/ 3 levels "female","male",..: 1 1 2 1 2 1 2 1 1 1 ...

$
County : Factor w/ 2 levels "Bear Lake","Elwood":
2 1 2 2 2 1 2 2 2 2 ...

$ Education : Factor w/ 5 levels "AA","BA","HS",..:
2 5 2 4 3 4 3 2 2 2 ...

$ Ed.code : int 3
5 3 4 1 4 1 3 3 3 ...

summary(Data)

Date
Instructor Student
Sex County

2015-11-01:10 Arthur Read :10
a : 1 female:10
Bear Lake: 4

2015-12-01: 7 Buster Baxter:
7 b : 1 male
: 6 Elwood :13

c : 1 other : 1

d : 1

e
: 1

f : 1

(Other):11

Education Ed.code

AA :2 Min. :1.000

BA
:7 1st Qu.:3.000

HS :2
Median :3.000

MA :4 Mean
:3.118

PHD:2 3rd Qu.:4.000

Max.
:5.00

### Remove
unnecessary objects

rm(Input)

##### Optional code to assign values to a variable based on another variable

Data$Ed.code[Data$Education=="HS"] = 1

Data$Ed.code[Data$Education=="AA"] = 2

Data$Ed.code[Data$Education=="BA"] = 3

Data$Ed.code[Data$Education=="MA"] = 4

Data$Ed.code[Data$Education=="PHD"] = 5

##### Optional code to change a factor variable to an ordered factor variable

Data$Education.ordered = factor(Data$Education,

ordered = TRUE,

levels = c("HS", "AA",
"BA", "MA", "PHD")

)

str(Data$Education.ordered)

summary(Data$Education.ordered)

HS AA BA MA PHD

2 2 7 4 2

median(Data$Education.ordered)

[1] "BA"

#### Statistics of location for ordinal data

Using the mean is usually not appropriate for ordinal
data. Instead the median should be used as a measure of location.
We will use our numerically-coded variable *Ed.code* for the analysis.

median(Data$Ed.code)

3

### Remember that 3 meant Bachelor’s degree in our coding

#### Statistics of variation for ordinal data

Statistics like standard deviation and standard error of the mean are usually inappropriate for ordinal data. Instead the five-number summary can be used.

summary(Data$Ed.code)

Min. 1st Qu. Median
Mean 3rd Qu. Max.

1.000 3.000
3.000 3.118 4.000 5.000

### Remember to ignore the
mean value for ordinal data.

### Remember that 1
meant High school, 3 meant Bachelor’s,

###
4 meant Master’s, and 5 meant Ph.D.

#### Statistics for grouped ordinal data

Because the *Summarize* function in the *
FSA* package reports the five-number summary, it is a useful function
for descriptive statistics for grouped ordinal data.

library(FSA)

Summarize(Ed.code ~ Sex,

data=Data)

Sex n nvalid mean
sd min Q1 median Q3 max percZero

1 female
10 10 3.5 0.9718253 2 3.00
3.0 4.00 5 0

2 male 6 6 2.5
1.3784049 1 1.25 2.5 3.75
4 0

3 other
1 1 3.0
NA 3 3.00 3.0 3.00 3
0

### Remember
to ignore the mean and sd values for ordinal data.

library(FSA)

Summarize(Ed.code ~ Sex + County,

data=Data)

Sex County
n nvalid mean
sd min Q1 median Q3 max percZero

1 female Bear Lake 3
3 4.666667 0.5773503 4 4.5
5 5 5 0

2 male Bear Lake 1 1 4.000000
NA 4 4.0 4 4
4 0

3 female
Elwood 7 7 3.000000 0.5773503
2 3.0 3 3 4
0

4 male Elwood 5
5 2.200000 1.3038405 1 1.0
2 3 4 0

5 other Elwood 1
1 3.000000 NA
3 3.0 3 3 3
0

### Remember
to ignore the mean and sd values for ordinal data.

#### Statistics for ordinal data treated as nominal data

The *summary* function in the native *base*
package and the *xtabs* function in the native *stats* package provide
counts for levels of a nominal variable.

The *prop.table* function translates a table into
proportions. The *margin=1* option indicates that the proportions are
calculated for each row.

First we will order the levels of *Education*, otherwise
R with report results in alphabetical order.

Data$Education = factor(Data$Education,

levels = c("HS", "AA",
"BA", "MA", "PHD"))

### Order factors, otherwise R will alphabetize
them

summary(Data$Education)

HS AA BA MA PHD

2 2 7 4 2

### Counts of each level of
Education

XT = xtabs( ~ Education,

data=Data)

XT

Education

HS AA BA MA PHD

2 2 7 4 2

prop.table(XT)

Education

HS AA BA MA PHD

0.1176471 0.1176471 0.4117647 0.2352941 0.1176471

##### Grouped data

XT = xtabs( ~ Sex + Education,

data = Data)

XT

Education

Sex HS AA BA MA PHD

female 0 1 5 2 2

male 2 1 1 2 0

other 0 0 1 0 0

prop.table(XT,

margin = 1)

Education

Sex HS AA BA MA PHD

female 0.0000000 0.1000000 0.5000000 0.2000000 0.2000000

male 0.3333333 0.1666667 0.1666667 0.3333333 0.0000000

other 0.0000000 0.0000000 1.0000000 0.0000000 0.0000000

### Proportion of responses for each row

##### Two-way grouped data

XT = xtabs(~ Sex + Education + County,

data=Data)

XT

### Note that the dependent variable,
Education, is the middle

### of the variable list.

, , County = Bear Lake

Education

Sex HS AA BA MA PHD

female 0 0 0 1 2

male 0 0 0 1 0

other 0 0 0 0 0

, , County = Elwood

Education

Sex HS AA BA MA PHD

female 0 1 5 1 0

male 2 1 1 1 0

other 0 0 1 0 0

### Descriptive statistics for nominal data

Descriptive statistics for nominal data consist
of listing or plotting counts for levels of the nominal data, often
by levels of a grouping variable. Tables of this information are
often called *contingency tables*.

For this example, we will look again at Arthur
and Buster’s data, but this time considering *Sex *to be the dependent
variable of interest.

If levels of a nominal variable are coded with
numbers, remember that the numbers will be arbitrary. For example,
if you assign *female* = 1, and *male* = 2, and *other*
= 3, it makes no sense to say that the average sex in Arthur’s class
was 1.3. Or that the average sex in Buster’s class was greater
than that in Arthur’s. It also makes no sense to say that the
median sex was female.

Input = ("

Date
Instructor Student Sex
County Education Ed.code

'2015-11-01' 'Arthur Read' a
female 'Elwood' BA
3

'2015-11-01' 'Arthur Read' b
female 'Bear Lake' PHD
5

'2015-11-01' 'Arthur Read' c
male 'Elwood' BA 3

'2015-11-01' 'Arthur Read' d
female 'Elwood' MA
4

'2015-11-01' 'Arthur Read' e
male 'Elwood' HS
1

'2015-11-01' 'Arthur Read' f
female 'Bear Lake' MA
4

'2015-11-01' 'Arthur Read' g
male 'Elwood' HS
1

'2015-11-01' 'Arthur Read' h
female 'Elwood' BA
3

'2015-11-01' 'Arthur Read' i
female 'Elwood' BA
3

'2015-11-01' 'Arthur Read' j
female 'Elwood' BA
3

'2015-12-01' 'Buster Baxter' k
male 'Elwood' MA
4

'2015-12-01' 'Buster Baxter' l
male 'Bear Lake' MA
4

'2015-12-01' 'Buster Baxter' m
female 'Elwood' AA
2

'2015-12-01' 'Buster Baxter' n
male 'Elwood' AA
2

'2015-12-01' 'Buster Baxter' o
other 'Elwood' BA
3

'2015-12-01' 'Buster Baxter' p
female 'Elwood' BA
3

'2015-12-01' 'Buster Baxter' q
female 'Bear Lake' PHD
5

")

Data = read.table(textConnection(Input),header=TRUE)

### Check the data
frame

Data

str(Data)

summary(Data)

###
Remove unnecessary objects

rm(Input)

#### Example of descriptive statistics for nominal data

The *summary* function in the native *base*
package and the *xtabs* function in the native *stats* package provide
counts for levels of a nominal variable.

The *prop.table* function translates a table into
proportions. The *margin=1* option indicates that the proportions are
calculated for each row.

##### One-sample data

summary(Data$Sex)

female male other

10 6 1

### Counts of each level of Sex

##### One-way data

xtabs(~ Date + Sex,

data=Data)

Sex

Date female male other

2015-11-01 7 3 0

2015-12-01 3 3 1

XT = xtabs(~ Date + Sex,

data=Data)

prop.table(XT,

margin = 1)

Sex

Date female male other

2015-11-01 0.7000000 0.3000000 0.0000000

2015-12-01 0.4285714 0.4285714 0.1428571

### Proportion of each level of Sex for each row

sum(XT)

[1] 17

### Sum of observation in the table

rowSums(XT)

2015-11-01 2015-12-01

10 7

### Sum of observation in each row of the table

colSums(XT)

female male other

10 6 1

### Sum of observation in each column of the table

##### Two-way data

xtabs(~ County + Sex + Date,

data=Data)

### Note that the dependent variable, Sex, is
the middle

### of the variable list.

, , Date = 2015-11-01

Sex

County female male other

Bear Lake 2 0 0

Elwood 5 3 0

, , Date = 2015-12-01

Sex

County female male other

Bear Lake 1 1 0

Elwood 2 2 1

#### Levels for factor variables

Most of the time in R, nominal variables will be handled by the software as factor variables.

The order of levels of factor variables are important because most functions, including plotting functions, will handle levels of the factor in order. The order of the levels can be changed, for example, to change the order that groups are plotted in a plot, or which groups are at the top of the table.

By default, the *read.table* function in R interprets
character data as factor variables. And by default R alphabetizes the levels
of the factors.

Looking at the Arthur and Buster data, note that *Instructor*,
*Student*, *Sex*, among other variables, are treated as factor
variables.

str(Data)

'data.frame': 17 obs. of 7 variables:

$ Date : Factor w/ 2 levels "2015-11-01","2015-12-01":
1 1 1 1 1 1 1 1

$ Instructor: Factor w/ 2 levels "Arthur Read",..: 1 1 1 1 1 1 1 1 1
1 ...

$ Student : Factor w/ 17 levels "a","b","c","d",..:
1 2 3 4 5 6 7 8 9 10 .

$ Sex : Factor w/ 3 levels "female","male",..: 1 1 2
1 2 1 2 1 1 1 ...

$ County : Factor w/ 2 levels "Bear Lake","Elwood": 2 1
2 2 2 1 2 2 2 2 .

$ Education : Factor w/ 5 levels
"AA","BA","HS",..: 2 5 2 4 3 4 3 2 2 2 ...

$ Ed.code : int 3 5 3 4 1 4 1 3 3 3 ...

Note also that the levels of the factor variables were alphabetized
by default. That is, even though *Elwood* was found in the data before *Bear
Lake*, R treats *Bear Lake* as the first level in the variable *County*.

summary(Data)

Date Instructor Sex County

2015-11-01:10 Arthur Read :10 female:10 Bear Lake: 4

2015-12-01: 7 Buster Baxter: 7 male : 6 Elwood :13

other : 1

We can order factor levels by the order in which they were read in the data frame.

Data$County = factor(Data$County,

levels=unique(Data$County))

levels(Data$County)

[1] "Elwood" "Bear Lake"

We can also order factor levels in an order of our choosing.

Data$Sex = factor(Data$Sex,

levels=c("male" , "female"))

levels(Data$Sex)

[1] "male" "female"

Note that in the actions above, we are not changing the order in the data frame, simply which level is treated internally by the software as "1" or "2", and so on.

### Required readings

*[Video]*** “Understanding
Summary statistics: Mean, Median, Mode”** from Statistics Learning
Center (Dr. Nic). 2015.
www.youtube.com/watch?v=rAN6DBctgJ0.

### Optional readings

__Statistics of location: mean and median__

** “Statistics of central tendency”** in
McDonald, J.H. 2014.

*Handbook of Biological Statistics*. www.biostathandbook.com/central.html.

** "The median outclasses the mean"** from
Dr. Nic. 2013. Learn and Teach Statistics & Operations Research.
learnandteachstatistics.wordpress.com/2013/04/29/median/.

*“Measures of the Location of the Data”,*** Section 2.3 **in Openstax. 2013.

*Introductory Statistics*. openstaxcollege.org/textbooks/introductory-statistics.

*“Measures of the Center of the Data”,*** Section 2.5 **in Openstax. 2013.

*Introductory Statistics*. openstaxcollege.org/textbooks/introductory-statistics.

*“Skewness and the Mean, Median, and Mode”,*** Section 2.6 **in Openstax. 2013.

*Introductory Statistics*. openstaxcollege.org/textbooks/introductory-statistics.

__Statistics of variation: standard deviation
and range__

** “Statistics of dispersion”** in McDonald,
J.H. 2014.

*Handbook of Biological Statistics*. www.biostathandbook.com/dispersion.html.

** “Standard error of the mean”** in McDonald,
J.H. 2014.

*Handbook of Biological Statistics*. www.biostathandbook.com/standarderror.html.

*“Measures of the Spread of the Data”,*** Section 2.7 **in Openstax. 2013.

*Introductory Statistics*. openstaxcollege.org/textbooks/introductory-statistics.

__The normal distribution__

*[Video]*** “Normal Distribution”**
from Statistics Learning Center (Dr. Nic). 2016.
www.youtube.com/watch?v=mtH1fmUVkfE.

** "The normal distribution – three tricky bits"**
from Dr. Nic. 2016. Learn and Teach Statistics & Operations Research.
learnandteachstatistics.wordpress.com/2016/01/18/the-normal-distribution/.

### Optional note on reporting summary statistics honestly

Every time we report a descriptive statistic or the result of a statistical test, we are condensing information, which may have been a whole data set, into one or a few pieces of information. It is the job of the analyst to choose the best way to present descriptive and graphical information, and to choose the correct statistical test or method.

##### SAT score example

Imagine a set of changes in SAT scores from seven
students who took a certain SAT preparation course. The change
in scores, *Increase*, represents the difference in score, that
is the score after taking the course minus their initial score.

Increase = c(50, 60, 120, -80, -10, 10, 0)

Our first instinct in assessing the course might be to look at the average (mean) change in score. The result is a mean increase of 21 points, which could be considered a success, perhaps.

mean(Increase)

[1] 21.42857

But we would be remiss to not look at other summary statistics and plots.

Using the *Summarize* function in the *FSA*
package, we find that the median increase was only 10 points.
The increase for the first quartile was negative, suggesting at least
25% of students got a lower score afterwards.

library(FSA)

Summarize(Increase,

digits = 2)

n
nvalid mean
sd min
Q1 median Q3
max

7.00 7.00
21.43 63.09 -80.00 -5.00
10.00 55.00 120.00

A histogram of the changes in scores suggests a slight right skew to the data, but that the mass of the data sits not far from zero.

hist(Increase,

col="gray")

Finally, we’ll compute the 95% confidence intervals for the mean change in score. Looking at the results for the percentile method, the confidence interval includes zero, suggesting the change in scores for this course were not statistically different from zero.

library(rcompanion)

Data = data.frame(Increase)

groupwiseMean(Increase ~ 1,

data = Data,

traditional = FALSE,

percentile = TRUE)

.id n Mean Conf.level Percentile.lower Percentile.upper

1 <NA> 7 21.4 0.95 -21.4 65.7

Based on the data exploration, it seems that it would be irresponsible or dishonest to simply report that the average increase in test scores was 20 points.

For descriptive statistics, we might report the mean and 95% confidence interval. Or perhaps the 5-point summary for the change in scores, or show the histogram of values.

### Optional analyses

#### Robust estimators: trimmed mean and Winsorized mean

Robust estimators of central tendency are used to describe the location of a data set without undue influence of extreme values. Robust estimators include trimmed means, Winsorized means, and other estimates like M-estimators (not discussed here).

##### Example

The New Jersey Envirothon is a statewide competition for high school students covering various aspects of natural resources science. In the Team Presentation station, student teams are judged by five judges.

Currently, scorekeepers drop the highest and lowest scores for each team to avoid the effect of aberrant low or high scores. This is an example of a trimmed mean. In this case, the mean is trimmed to 60% (with 20% removed from each side).

Another option to ameliorate the effect of extreme scores is using Winsorized means. A Winsorized mean removes extreme observations, but replaces them with the closest observations in terms of magnitude.

In this example, two teams received five scores for their presentations that have the same mean and median. We will look at robust estimators to determine if one team should be scored higher than the other.

Team.A = c(100, 90, 80, 60, 20)

median(Team.A)

[1] 80

mean(Team.A)

[1] 70

mean(Team.A,

trim = .20)
# This trims to the inner 60% of
observations

[1] 76.66667

library(psych)

winsor(Team.A,

trim = 0.20)
# This Winsorizes to the inner
60% of observations

[1] 92 90 80 60 52

### Note that the Winsorized
values at the extremes appear to be calculated

### with a function analogous to the quantile(x, probs,
type = 7) function.

winsor.mean(Team.A,

trim = 0.20) # This
Winsorizes to the inner 60% of observations

[1] 74.8

Team.B = c(80, 80, 80, 80, 30)

median(Team.B)

[1] 80

mean(Team.B)

[1] 70

mean(Team.B,

trim = .20)
# This trims to the inner 60% of
observations

[1] 80

library(psych)

winsor(Team.B,

trim = 0.20)
# This Winsorizes to the inner
60% of observations

[1] 80 80 80 80 70

### Note that the Winsorized
values at the extremes appear to be calculated

### with a function analogous to the quantile(x, probs,
type = 7) function.

winsor.mean(Team.B,

trim = 0.20) # This Winsorizes to the inner 60% of observations

[1] 78

In this example, the means and medians for Team A and Team B were identical. However, the trimmed mean for Team A was less than for Team B (77 vs. 80), and the Winsorized mean for A was less than for Team B (75 vs. 78). According to either the trimmed mean method or the Winsorized mean method, Team B had a higher score.

Note also that the Winsorized means were lower than for the trimmed means, for both teams. This is because the Winsorized mean is better able to take into account the low scores for each team.

#### Geometric mean

The geometric mean is used to summarize certain measurements, such as average return for investments, and for certain scientific measurements, such as bacteria counts in environmental water. It is useful when data are log-normally distributed.

Practically speaking, using the geometric mean ameliorates the effect of outlying values in the data.

To get the geometric mean, the log of each value is taken, these values are averaged, and then the result is the base of the log raised to this value.

Imagine a series of 9 counts of bacteria from lake
water samples, called *Bacteria* here. The geometric mean
can be calculated with a nested *log*, *mean*, and *exp*
functions. Or more simply, the *geometric.mean* function
in the *psych* package can be used.

##### Bacteria example

Bacteria = c(20, 40, 50, 60, 100, 120, 150, 200, 1000)

exp(mean(log(Bacteria)))

[1] 98.38887

library(psych)

geometric.mean(Bacteria)

[1] 98.38887

hist(Bacteria,
### Produce a histogram of values

col="darkgray",

breaks="FD")

##### Annual return example

Geometric means are also used to calculate the average annual return for investments. Each value in this vector represents the percent return of the investment each year. The arithmetic mean will not give the correct answer for average annual return.

Return = c(-0.80, 0.20, 0.30, 0.30, 0.20, 0.30)

library(psych)

geometric.mean(Return + 1)-1

[1] -0.07344666

### The geometric mean will
give the correct answer for average

### annual
return

This can also be calculated manually. If you start with 100 dollars with this investment, you will end up with 63 dollars, equivalent to a return of – 0.07.

100 * (1-0.80) * 1.20 * 1.30 * 1.30 * 1.20 * 1.30

[1] 63.2736

#### Harmonic mean

The harmonic mean is another type of average that is used in certain situations, such as averaging rates or speeds.

To get the harmonic mean, the inverse of each value is taken, these values are averaged, and then the inverse of this result is reported.

Imagine a series of 9 speeds, called *Speed*
here. The harmonic mean can be calculated with a nested *1/x*,
*mean*, and *1/x* functions. Or more simply, the *
harmonic.mean* function in the *psych* package can be used.

Speed = c(20, 40, 50, 60, 100, 120, 150, 200, 1000)

1/mean(1/Speed)

63.08411

library(psych)

harmonic.mean(Speed)

63.08411

### Exercises C

1. Considering Ren and Stimpy's workshops together,

a. How many total attendees were there?

b. How many total workshops were there?

c. How many locations were there?

d. What was the mean number of attendees?

e. What was the median number of attendees?

2. Considering Ren and Stimpy's workshops,

a. Which workshops had a higher mean number of attendees and
what was the mean? Ren’s or Stimpy’s?

b. Which workshops had a higher mean number of attendees and what was the mean? Ren North or Ren South?

3. Considering Arthur and Buster's workshops together,

a How many students attended in total?

b. How many distinct levels of education were reported?

c. What was the median level of education reported?

d. 75% of students had what level of education or lower?

e. What was the median education level for females?

f. What was the median education level for males?

g. What was the median education level for females from Elwood
County?

h. What was the median education level for males from Elwood County?

4. Considering Arthur and Buster's workshops,

a. How many students were male?

b. What percentage is this of total students?

c. How many students in the November workshop were male?

d. What percentage is this of total students in the November
workshop?

e. How many male students in the November workshop were from Bear Lake County?

5. As part of a nutrition education program, extension
educators had students keep diaries of what they ate for a day and then
calculated the calories students consumed. The following data are the result.
*Rating* indicates the score, from 1 to 5, that students gave as to the
usefulness of the program. It should be considered an ordinal variable.

Student Teacher Sex Calories Rating

a Tetsuo male 2300 3

b Tetsuo female 1800 3

c Tetsuo male 1900 4

d Tetsuo female 1700 5

e Tetsuo male 2200 4

f Tetsuo female 1600 3

g Tetsuo male 1800 3

h Tetsuo female 2000 3

i Kaneda male 2100 4

j Kaneda female 1900 5

k Kaneda male 1900 4

l Kaneda female 1600 4

m Kaneda male 2000 4

n Kaneda female 2000 5

o Kaneda male 2100 3

p Kaneda female 1800 4

a. What are the variables in this data set and what type of variable is each? (Types are “nominal”, “ordinal”, and “interval/ratio”, not how their types are reported by R.)

For each of the following, answer the question, and ** show
the output from the analyses you used to answer the question**.

b. How many students were involved in this data set?

c. What was the mean caloric intake?

d. What was the median caloric intake?

e. What was the standard deviation of caloric intake?

f. What was the mean caloric intake for females?

g. What was the mean caloric intake for females in Kaneda’s class?

h. How many males are in Kaneda’s class?

6. What do you conclude in practical terms about the caloric intake data for
Tetsuo’s and Kaneda’s classes? You might consider the mean or median values
between males and females or between Tetsuo and Kaneda. Think about the real
world. Are the differences you are looking at important in a practical sense?

Be sure to pay attention to the spread of data as you think about this, for example using minimums and maximums, Q1 and Q3 values, or standard deviation values.

Be quantitative in your answer. That is, if you are talking about a difference in medians, mention the actual numerical difference. If expressing values as a percentage makes sense, feel free to do so.