Describing Distributions with Numbers
Diana Mindrila, Ph.D.
Phoebe Balentyne, M.Ed.
Based on Chapter 2 of The Basic Practice of Statistics (6
th
ed.)
Concepts:
Measuring Center: Mean and Median
Measuring Spread: Quartiles
Five-Number Summary and Boxplots
Spotting Suspected Outliers
Measuring Spread: Standard Deviation
Choosing Measures of Center and Spread
Objectives:
Calculate and interpret mean, median, and mode
Compare mean and median
Calculate and interpret quartiles
Construct and interpret the five-number summary and boxplots
Determine suspected outliers
Calculate and interpret standard deviation
Choose appropriate measures of center and spread
Organize a statistical problem
References:
Moore, D. S., Notz, W. I, & Flinger, M. A. (2013). The basic practice of statistics (6
th
ed.). New York, NY: W. H. Freeman and Company.
Measuring Center: The Mean
The most common measure of center is the arithmetic average, or mean.
The mean of a data set is the average of all its values.
Calculate the mean by finding the sum of all the values, then dividing by the
number of values in the data set.
The mean is a measure of the center, because it is an indicator of where most
values are located.
Notation for mean: 𝑥̅ (spoken: x bar)
The mean is not resistant to outliers.
To find the mean 𝑥 (pronounced “x-bar”) of a set of observations, add their values and
divide by the number of observations. If the n observations are x
1
, x
2
, x
3
, …, x
n
, their mean
is:
or, in more compact notation,
x =
sum of observations
n
=
x
1
+ x
2
+ ...+ x
n
n
x =
x
i
å
n
StudentID
TestScore
1
91
2
89
3
89
4
88
5
95
6
100
7
94
8
30
9
92
10
93
Example of Calculating the Mean:
Entire Sample:
Sum = 861
Number of Students: N = 10
Mean = Sum/N
Mean = 861/10 -> Mean = 86.1
Sample with outlier removed:
Sum = 831
N = 9
Mean = 92.33
Calculate the mean:
Find the sum: 91+89+89+88+95+100+94+30+92+93 = 861
Divide by the number of values: 861/10
Mean = 86.1
Looking at the data after calculating the mean shows that all but one student
(student 8) scored above the mean. Even without a graph, it is evident that
this score does not follow the general pattern; it is an outlier.
Remove the outlier and recalculate the mean:
Find the sum: 91+89+89+88+95+100+94+92+93 = 861
Divide by the number of values: 831/9
Mean = 92.33
There is a considerable difference between the values of the mean with and
without the outlier.
As this example portrays, the mean is considered a nonresistant measure,
because outliers can have a strong impact on its value.
This is why it is important to use additional descriptive statistics to inform
our decisions, not just the mean.
Measuring Center: The Median
Because the mean cannot resist the influence of extreme observations, it is not a
resistant measure of center. Another common measure of center is the median.
Middle Point
Odd # of Observations: (N+1)/2 [Use only one value]
Example: (9+1)/2 = 5 > the median is the value of the 5
th
observation
Even # of Observations: (N+1)/2 [Use two consecutive values]
Example: (10+1)/2 = 5.5 > find the midpoint (or average) of the 5
th
and 6
th
observations
The median is a summary statistic that indicates the midpoint of a
distribution.
The following general rule can be helpful:
If the data set contains an odd number of observations, like 9, add one to
this number and get 10. Then, divide 10 by 2, and obtain five, which
means that the fifth observation is the median. In this case, only this
value is used and an average does not need to be computed.
If the data set contains an even number of observations, like 10, add 1 to
this number and get 11. Then, divide 11 by 2, and obtain 5.5. This means
that the fifth and sixth observations should be used to compute the
median.
Statistical software computes the median, so it is not necessary to do it by
hand, but it is important to understand how it is obtained to be able to
understand what it means and how to interpret its values.
The median M is the midpoint of a distribution, the number such that half of the
observations are smaller and the other half are larger.
To find the median of a distribution:
1. Arrange all observations from smallest to largest.
2. If the number of observations n is odd, the median M
is the center observation
in the ordered list.
3. If the number of observations n is even, the median M is the average of the
two center observations in the ordered list.
StudentID
TestScore
4
88
2
89
3
89
1
91
9
92
10
93
7
94
5
95
6
100
Example of Finding the Median:
Median: (91+92)/2 = 91.5
The median can be found using a table of values or a stem-and-leaf plot. In a
stem-and-leaf plot, the values are already presented in ascending order.
In the above example, the middle values have been highlighted. Since there
are two middle values, they were averaged to find the median.
Example with Outlier Removed:
Median = 92
When there is only one middle value, it is the median. No calculation is
necessary.
It is important to note that the median value without the outlier is very close
to the median with the outlier. Outliers do not have a strong impact on the
median.
The median is not as sensitive to outliers as the mean, so it is important to
always report both the mean and the median, especially when data are
skewed.
StudentID
TestScore
8
30
4
88
2
89
3
89
1
91
9
92
10
93
7
94
5
95
6
100
Stem
Leaf
3
0
4
5
6
7
8
899
9
12345
10
0
Stem
Leaf
8
899
9
12345
10
0
StudentID
TestScore
8
30
4
88
2
89
3
89
1
91
9
92
10
93
7
94
5
95
6
100
Measuring Center: The Mode
The mode is the value that occurs most often in a data set.
Example:
Mode = 89
In this example, 89 is the only value that occurs more than once. It occurs
twice. It has been highlighted in red. This value is the mode.
Comparing the Mean and Median
The mean and median measure center in different ways, and both are useful.
The mean and median of a roughly symmetric distribution are
close together.
If the distribution is exactly
symmetric
, the mean and median are
exactly the same.
In a
skewed distribution, the mean is usually farther out in the
long tail than the median.
Symmetric Distribution
Mean, Median, Mode:
approximately equal
Positively Skewed Distribution
Mean larger than median
Negatively Skewed Distribution
Mean smaller than median
Figure A displays a symmetric distribution. The mean, median, and mode
are all approximately equal. With real data, these will not have they exact
same value, but they will be very close.
Outliers cause a skewed distribution resulting in a larger difference between
the mean and median. When there are extreme observations, the mean is
pulled in that direction.
The mode is the value that occurs most often, so it is always located at the
highest point in the curve.
If the outliers in a data set are larger values, then the distribution will be
positively skewed, as shown in Figure B. In this case, the mean has been
pulled to the right with the extremely large values. The mean is larger than
the median in a positively skewed distribution.
If the outliers in a data set are smaller values, then the distribution will be
negatively skewed, as shown in Figure C. In this case, the mean has been
pulled to the left with the extremely small values. The mean is smaller than
the median in a negatively skewed distribution.
StudentID
TestScore
1
91
2
89
3
89
4
88
5
95
6
100
7
94
8
30
9
92
10
93
Measuring Spread: Range
The range of a data set is the difference between the minimum and maximum
recorded values.
Example of Calculating the Range:
Min = 30 Max = 100
Range = Max – Min
Range = 100 – 30
Range = 70
Calculate the range: Range = Maximum Value – Minimum Value
Measuring Spread: Quartiles
A measure of center alone can be misleading. A useful numerical description of a
distribution requires both a measure of center and a measure of spread.
Mean, median, mode, and range are helpful measures, but they are not
enough.
To provide even more information, the quartiles and interquartile range
should be identified.
To calculate the quartiles:
1) Arrange the observations in increasing order and locate
the median M.
2) The first quartile Q
1
is the median of the observations
located to the left of the median in the ordered list.
3) The third quartile Q
3
is the median of the observations
located to the right of the median in the ordered list.
The
interquartile range (IQR) is defined as:
IQR = Q
3
– Q
1
How to Calculate the Quartiles and the Interquartile Range
StudentID
TestScore
8
30
Min
4
88
2
89
Q1=89
3
89
1
91
Median=91.5
(Q2)
9
92
10
93
7
94
Q3=94
5
95
6
100
Max(Q4)
Example:
Quartiles get their name because they each represent a quarter, or 25%, of
the values in the data set.
The first quartile, Q
1
, is found by finding the median of the values between
the minimum and the median. In this case, the values in the first half of the
data set are: 30, 88, 89, 89, and 91. The median of these values: Q
1
= 89
The median would be the second quartile, Q
2
, but should be simply referred
to as the median.
The third quartile, Q
3
, is found by finding the median of the values between
the median and the maximum. In this case, the values in the second half of
the data set are: 92, 93, 94, 95, and 100. The median of these values: Q
3
= 94
Each quartile contains 25% of the values in the data set, so 50% of the values
are between Q
1
and Q
3
. The range represents the spread of the entire data
set.
The interquartile range represents the spread of the middle 50% of values.
This value is found with the following equation: IQR = Q
3
– Q
1
In this example, IQR = Q
3
– Q
1
= 94 – 89 = 5
The value of Q
3
in this example is very close to the maximum. However, the
value of Q
1
is very far from the minimum. This indicates that the distribution
is skewed to the left.
25%
25%
25%
25%
IQR = 94 – 89
IQR = 5
(50% of values)
Five-Number Summary
The maximum and minimum values alone do not give enough information
about the distribution as a whole.
Likewise, the median and the quartiles do not give information about the
tails of the distribution.
The five-number summary combines all of these values to get a quick
summary of both center and spread. It includes the minimum, the first
quartile (Q
1
), the median (Q
2
), the third quartile (Q
3
), and the maximum (Q
4
).
Example:
Min
Q1
Median(Q2)
Q3
Max(Q4)
30
89
91.5
94
100
The five-number summary of a distribution consists of the
smallest observation, the first quartile, the median, the third
quartile, and the largest observation, written in order from
smallest to largest.
Minimum Q
1
Median Q
3
Maximum
Boxplots
The five-number summary divides the distribution roughly into quarters.
This leads to a new way to display quantitative data: the boxplot.
The boxplot, or box-and-whisker plot, is a graphical representation of the
five-number summary.
The above images are examples of boxplots obtained using SPSS software.
The variable of interest is on the horizontal axis and the values that variable
takes are on the vertical axis.
The box represents the interquartile range, or the values that are located
between the first and third quartiles, so the box includes 50% of the sample.
The line across the box displays the location of the median value. The
whiskers extend from the box to the minimum and the maximum values. The
whiskers help determine whether or not the data are skewed. If they have
approximately the same length, the variable has a symmetric distribution. In
this example, the lower whisker is much longer, so the data are negatively
skewed.
Boxplots sometimes represent the outliers as dots or little stars outside of
the whiskers. In this way, extreme observations are identified, but are not
considered as being the actual minimum or maximum values. The example
on the right represents the data set from the previous example, but the
individual with a score of 30 is considered an outlier, and the whisker starts
from the next lowest value, which is 88.
In the boxplot on the right, where the outlier has been removed, each section
of the box is approximately the same size and the whiskers are
maximum
Q
3
median
Q
1
minimum
approximately the same length. This represents a more symmetric
distribution.
Stem
Leaf
3
0
4
5
6
7
8
899
9
12345
10
0
Suspected Outliers: The 1.5 X IQR Rule
In addition to serving as a measure of spread, the interquartile range (IQR) is
also used as part of a rule for identifying outliers.
A researcher may need to make a decision as to whether or not a data point
is an outlier.
The 1.5 X IQR Rule can be used as a rule of thumb to test suspected outliers.
To test outliers:
o Multiply the IQR by 1.5
o Check for small outliers by subtracting (IQR X 1.5) from Q
1
. Any
values below that number would be considered outliers.
o Check for large outliers by adding (IQR x 1.5) to Q
3
. Any values above
that number would be considered outliers.
This rule helps identifying outliers to be less arbitrary.
Example:
Q
1
= 89 Q
3
= 94 IQR = 5
1.5 x IQR = 1.5 X 5 = 7.5
Q
1
– 1.5 X IQR = 89 – 7.5 = 81.5
Q
3
+ 1.5 X IQR = 94 + 7.5 = 101.5
**Any score lower than 81.5 or higher than 101.5
is considered an outlier.
In this example:
o Values lower than 81.5 would be considered outliers.
o Values greater than 101.5 would be considered outliers.
Using this rule, the value of 30 can be considered an outlier.
The 1.5 IQR Rule for Outliers
An observation is considered an outlier if it falls more than
1.5 IQR above the third quartile or below the first quartile.
Measuring Spread: Standard Deviation
The most common measure of spread looks at how far each observation is
from the mean. This measure is called the standard deviation.
To find the standard deviation:
o Compute the mean of all values.
o Find the difference between each value in the data set and the mean.
(Find out how far each value is from the mean).
o Square each of these differences. (This will make all the values
positive).
o Calculate the sum of these values (the squares).
o Divide the sum by (N – 1) where N is the number of values in the data
set. (N – 1 is also called the degrees of freedom for the data set).
o The value obtained is the variance, which shows how much spread
there is in the data.
o To obtain the standard deviation take the square root of the
variance.
The variance and standard deviation are rarely computed by hand. This task
can become extremely cumbersome with large data sets.
Variance and standard deviation are easily obtained using statistical
software, but it is important to understand their meaning.
The standard deviation is simply the average distance from the mean.
o A large standard deviation means the spread is large.
o A small standard deviation means the values in the data set are
clustered around the mean.
Like the mean, the standard deviation is very sensitive to outliers. If only one
difference is very large, the overall average distance from the mean increases
significantly.
The standard deviation s
x
measures the average distance of
the observations from their mean. It is calculated by finding an
average of the squared distances, and then taking the square
root. This average squared distance is called the
variance.
variance = s
x
2
=
(x
1
- x )
2
+ (x
2
- x )
2
+ ...+ (x
n
- x )
2
n -1
=
1
n -1
(x
i
- x )
2
å
standard deviation = s
x
=
1
n -1
(x
i
- x )
2
å
Calculating the Standard Deviation
Example:
The mean in the above example is 86.1.
First, the distance from the mean was computed for each test score by
subtracting each value from the mean. These differences are displayed in the
third column. Some of these differences are negative, because all of the
values except 30 (the outlier) are larger than the mean.
Second, each of the distances was squared. These values are displayed in the
fourth column.
Next, the variance was computed by finding the sum of the squares (3608.9)
and dividing it by N – 1. Since there are 10 values in this data set:
N – 1 = 10 – 1 = 9. Therefore, the variance = 3608.9/9 = 400.99
Finally, the standard deviation was obtained by taking the square root of the
variance. (20.02)
StudentID
TestScore
Distancefromt hemean
Squared
Distance
1
91
-4.9
24.01
2
89
-2.9
8.41
3
89
-2.9
8.41
4
88
-1.9
3.61
5
95
-8.9
79.21
6
100
-13.9
193.21
7
94
-7.9
62.41
8
30
56.1
3147.21
9
92
-5.9
34.81
10
93
-6.9
47.61
Mean
86.1
Sum
3608.9
Variance
(Averageo fSquaredDistances)
400.99
StandardDeviation
(SquareRootoftheVariance)
20.02
Choosing Measures of Center and Spread
Two choices for descriptions of center and spread have been discussed thus
far:
o Mean and Standard Deviation
o Median and Interquartile Range
Depending on the purpose of a study and the nature of the data, some
measures of center and spread are more appropriate or more informative
than others.
The mean and standard deviation are sensitive to outliers. Therefore,
especially when the data are skewed, it is important to also report the
median and the interquartile range and to explain their meaning.
Usually, it is helpful to report all of these numbers and explain the
significance of each one.
Choosing Measures of Center and Spread
• The median and IQR are usually better than the mean and
standard deviation for describing a skewed distribution or a
distribution with outliers.
•
Use mean and standard deviation only for reasonably
symmetric distributions that don’t have outliers.
•
NOTE: Numerical summaries do not fully describe the
shape of a distribution. ALWAYS PLOT THE DATA!
Organizing a Statistical Problem
There is a four-step process that can be used to solve statistical problems as
they become more complex.
The purpose of the work of a researcher is to respond to questions based on
statistical evidence. In order to respond to questions and to provide
evidence, researchers must conduct some statistical analyses. This is a
complex process, and, in real life, it is often difficult to determine how to
begin.
The first thing that the researcher should do is to formulate a question. The
question relies on his or her knowledge and experience in the field and must
also take into account the constraints of reality.
The second step is to identify the statistical procedures that will help the
researcher respond to the question. Regardless of the statistical analysis that
is employed, the researcher must first screen the data, which means to
examine potential missing values, data entry errors, etc. Then, the
researcher must always examine the data using descriptive analyses.
Descriptive analyses can be presented using both graphs and numbers,
depending on the nature of the data. It is up to the researcher to choose the
descriptive analysis that is the most relevant to the study.
A researcher should always examine the data using descriptive analyses
prior to employing other statistical procedures and interpreting the results.
State: What’s the practical question, in the context of the real-
world setting?
Plan:
What specific statistical operations does this problem call
for?
Do:
Make graphs and carry out calculations needed for the
problem.
Conclude:
Give a practical conclusion in the setting of the real-
world problem.
How to Organize a Statistical Problem: A Four-Step Process
