Chapter 8: Descriptive Statistics
Introduction
A statistic is a numerical representation of information. Whenever we quantify or apply numbers
to data in order to organize, summarize, or better understand the information, we are using
statistical methods. These methods can range from somewhat simple computations such as
determining the mean of a distribution to very complex computations such as determining factors
or interaction effects within a complex data set. This chapter is designed to present an overview
of statistical methods in order to better understand research results. Very few formulas or
computations will be presented, as the goal is merely to understand statistical theory.
Before delving into theory, it is important to understand some basics of statistics. There are two
major branches of statistics, each with specific goals and specific formulas. The first, descriptive
statistics, refers to the analysis of data of an entire population. In other words, descriptive
statistics is merely using numbers to describe a known data set. The term population means we
are using the entire set of possible subjects as opposed to just a sample of these subjects. For
instance, the average test grade of a third grade class would be a descriptive statistic because
we are using all of the students in the class to determine a known average.
Second, inferential statistics, has two goals: (1) to determine what might be happening in a
population based on a sample of the population (often referred to as estimation) and (2) to
determine what might happen in the future (often referred to as prediction). Thus, the goals of
inferential statistics are to estimate and/or predict. To use inferential statistics, only a sample of
the population is needed. Descriptive statistics, however, require the entire population be used.
Many of the descriptive techniques are also used for inferential data so we’ll discuss these first.
Lets start with a brief summary of data quality.
Scales of Measurement
Statistical information, including numbers and sets of numbers, has specific qualities that are of
interest to researchers. These qualities, including magnitude, equal intervals, and absolute zero,
determine what scale of measurement is being used and therefore what statistical procedures
are best. Magnitude refers to the ability to know if one score is greater than, equal to, or less
than another score. Equal intervals means that the possible scores are each an equal distance
from each other. And finally, absolute zero refers to a point where none of the scale exists or
where a score of zero can be assigned.
When we combine these three scale qualities, we can determine that there are four scales of
measurement. The lowest level is the nominal scale, which represents only names and
therefore has none of the three qualities. A list of students in alphabetical order, a list of favorite
cartoon characters, or the names on an organizational chart would all be classified as nominal
data. The second level, called ordinal data, has magnitude only, and can be looked at as any
set of data that can be placed in order from greatest to lowest but where there is no absolute
zero and no equal intervals. Examples of this type of scale would include Likert Scales and the
Thurstone Technique.
The third type of scale is called an interval scale, and possesses both magnitude and equal
intervals, but no absolute zero. Temperature is a classic example of an interval scale because
we know that each degree is the same distance apart and we can easily tell if one temperature
is greater than, equal to, or less than another. Temperature, however, has no absolute zero
because there is (theoretically) no point where temperature does not exist.
Finally, the fourth and highest scale of measurement is called a ratio scale. A ratio scale
contains all three qualities and is often the scale that statisticians prefer because the data can be
more easily analyzed. Age, height, weight, and scores on a 100-point test would all be
examples of ratio scales. If you are 20 years old, you not only know that you are older than
someone who is 15 years old (magnitude) but you also know that you are five years older (equal
intervals). With a ratio scale, we also have a point where none of the scale exists; when a
person is born his or her age is zero.
Table 8.1: Scales of Measurement
Scale
Level
Scale of
Measurement
Scale
Qualities
Example(s)
Magnitude
4
Ratio
Equal
Intervals
Age, Height, Weight,
Percentage
Absolute
Zero
Magnitude
3
Interval
2
Ordinal
Magnitude
Likert Scale, Anything
rank ordered
1
Nominal
None
Names, Lists of words
Equal
Intervals
Temperature
Types of Distributions
When datasets are graphed they form a picture that can aid in the interpretation of the
information. The most commonly referred to type of distribution is called a normal distribution or
normal curve and is often referred to as the bell shaped curve because it looks like a bell. A
normal distribution is symmetrical, meaning the distribution and frequency of scores on the left
side matches the distribution and frequency of scores on the right side.
Many distributions fall on a normal curve, especially when large samples of data are considered.
These normal distributions include height, weight, IQ, SAT Scores, GRE and GMAT Scores,
among many others. This is important to understand because if a distribution is normal, there are
certain qualities that are consistent and help in quickly understanding the scores within the
distribution
The mean, median, and mode of a normal distribution are identical and fall exactly in the center of
the curve. This means that any score below the mean falls in the lower 50% of the distribution of
scores and any score above the mean falls in the upper 50%. Also, the shape of the curve allows
for a simple breakdown of sections. For instance, we know that 68% of the population fall
between one and two standard deviations (See Measures of Variability Below) from the mean
and that 95% of the population fall between two standard deviations from the mean. Figure 8.1
shows the percentage of scores that fall between each standard deviation.
Figure 8.1: The Normal Curve
As an example, lets look at the normal curve associated with IQ Scores (See Figure 8.2). The
mean, median, and mode of a Wechsler’s IQ Score is 100, which means that 50% of IQs fall at
100 or below and 50% fall at 100 or above. Since 68% of scores on a normal curve fall within
one standard deviation and since an IQ score has a standard deviation of 15, we know that 68%
of IQs fall between 85 and 115. Comparing the estimated percentages on the normal curve with
the IQ scores, you can determine the percentile rank of scores merely by looking at the normal
curve. For example, a person who scores at 115 performed better than 87% of the population,
meaning that a score of 115 falls at the 87th percentile. Add up the percentages below a score of
115 and you will see how this percentile rank was determined. See if you can find the percentile
rank of a score of 70.
Figure 8.2: IQ Score Distributions
Skew. The skew of a distribution refers to how the curve leans. When a curve has extreme
scores on the right hand side of the distribution, it is said to be positively skewed. In other words,
when high numbers are added to an otherwise normal distribution, the curve gets pulled in an
upward or positive direction. When the curve is pulled downward by extreme low scores, it is said
to be negatively skewed. The more skewed a distribution is, the more difficult it is to interpret.
Figure 8.3: Distribution Skew
Kurtosis. Kurtosis refers to the peakedness or flatness of a distribution. A normal distribution or
normal curve is considered a perfect mesokurtic distribution. Curves that contain more score in
the center than a normal curve tend to have higher peaks and are referred to as leptokurtic.
Curves that have fewer scores in the center than the normal curve and/or more scores on the
outer slopes of the curve are said to be platykurtic.
Figure 8.4: Distribution Kurtosis
Statistical procedures are designed specifically to be used with certain types of data, namely
parametric and non-parametric. Parametric data consists of any data set that is of the ratio or
interval type and which falls on a normally distributed curve. Non-parametric data consists of
ordinal or ratio data that may or may not fall on a normal curve. When evaluating which statistic
to use, it is important to keep this in mind. Using a parametric test (See Summary of Statistics in
the Appendices) on non-parametric data can result in inaccurate results because of the difference
in the quality of this data. Remember, in the ideal world, ratio, or at least interval data, is
preferred and the tests designed for parametric data such as this tend to be the most powerful.
Measures of Central Tendency
There are three measures of central tendency and each one plays a different role in determining
where the center of the distribution or the average score lies. First, the mean is often referred to
as the statistical average. To determine the mean of a distribution, all of the scores are added
together and the sum is then divided by the number of scores. The mean is the preferred
measure of central tendency because it is used more frequently in advanced statistical
procedures, however, it is also the most susceptible to extreme scores. For example, if the
scores ‘8’ ‘9’ and ‘10’ were added together and divided by ‘3’, the mean would equal ‘9’. If the 10
was changed to 100, making it an extreme score, the mean would change drastically. The new
mean of ‘8’ ‘9’ and ‘100’ would be ’39.’
The median is another method for determining central tendency and is the preferred method for
highly skewed distributions. The media is simply the middle most occurring score. For an even
number of scores there will be two middle numbers and these are simply added together and
divided by two in order to determine the median. Using the same distribution as above, the
scores ‘8’ ‘9’ and ‘10’ would have a median of 9. By changing the ‘10’ to a score of ‘100’ you’ll
notice that the median of this new positively skewed distribution does not change. The median
remains equal to ‘9.’
Finally, the mode is the least used measure of central tendency. The mode is simply the most
frequently occurring score. For distributions that have several peaks, the mode may be the
preferred measure. There is no limit to the number of modes in a distribution. If two scores tie as
the most frequently occurring score, the distribution would be considered bimodal. Three would
be trimodal, and all distributions with two or more modes would be considered multimodal
distributions.
Figure 8.5: Measures of Central Tendency
Interestingly, in a perfectly normal distribution, the mean, median, and mode are exactly the
same. As the skew of the distribution increases, the mean and median begin to get pulled toward
the extreme scores. The mean gets pulled the most which is why it becomes less valid the more
skewed the distribution. The median gets pulled a little and the mode typically remains the
same. You can often tell how skewed a distribution is by the distance between these three
measures of central tendency.
Measures of Variability
Variability refers to how spread apart the scores of the distribution are or how much the scores
vary from each other. There are four major measures of variability, including the range,
interquartile range, variance, and standard deviation. The range represents the difference
between the highest and lowest score in a distribution. It is rarely used because it considers only
the two extreme scores. The interquartile range, on the other hand, measures the difference
between the outermost scores in only the middle fifty percent of the scores. In other words, to
determine the interquartile range, the score at the 25th percentile is subtracted from the score at
the 75th percentile, representing the range of the middle 50 percent of scores.
The variance is the average of the squared differences of each score from the mean. To
calculate the variance, the difference between each score and the mean is squared and then
added together. This sum is then divided by the number of scores minus one. When the square
root is taken of the variance we call this new statistic the standard deviation. Since the variance
represents the squared differences, the standard deviation represents the true differences and is
therefore easier to interpret and much more commonly used. Since the standard deviation relies
on the mean of the distribution, however, it is also affected by extreme scores in a skewed
distribution.
The Correlation
The correlation is one of the easiest descriptive statistics to understand and possibly one of the
most widely used. The term correlation literally means co-relate and refers to the measurement of a
relationship between two or more variables. A correlational coefficient is used to represent this
relationship and is often abbreviated with the letter ‘r.’ A correlational coefficient typically ranges
between –1.0 and +1.0 and provides two important pieces of information regarding the relationship:
Intensity and Direction.
Intensity refers to the strength of the relationship and is expressed as a number between zero
(meaning no correlation) and one (meaning a perfect correlation). These two extremes are rare as
most correlations fall somewhere in between. In the social sciences, a correlation of 0.30 may be
considered significant and any correlation above 0.70 is almost always significant. The absolute
value of ‘r’ represents the intensity of any correlation.
Direction refers to how one variable moves in relation to the other. A positive correlation (or
direct relationship) means that two variables move in the same direction, either both moving up
or both moving down. For instance, high school grades and college grades are often positively
correlated in that students who earn high grades in high school tend to also earn high grades in
college. A negative correlation (or inverse relationship) means that the two variables move in
opposite directions; as one goes up, the other tends to go down. For instance, depression and
self-esteem tend to be inversely related because the more depressed an individual is the lower
his or her self-esteem. As depression increases, then, self-esteem tends to decrease. The sign
in front of the ‘r’ represents the direction of a correlation.
Figure 8.6: Scatter plots for sample correlations
Scatter Plot. Correlations are graphed on a special type of graph called a scatter plot (or
scatter gram). On a scatter plot, one variable (typically called the X variable) is placed on the
horizontal axis (abscissa) and the Y variable is placed on the vertical axis (ordinate). For
example, if we were measuring years of work experience and yearly income, we would likely find
a positive correlation. Imagine we looked at ten subjects and found the hypothetical results
listed in Table 8.2.
Table 8.2: Sample Correlation Data
Subject
Number
Experience
in Years
Income in
Thousands
Subject
Number
Experience
in Years
Income in
Thousands
1
0
20
6
15
50
2
5
30
7
20
60
3
5
40
8
25
50
4
10
30
9
30
70
5
10
50
10
35
60
Notice how each subject has two pieces of information (years of experience and income).
These are the two variables that we are looking at to determine if a relationship exists. To place
this information in a scatter plot we will consider experience the X variable and income the Y
variable (the results will be the same even if the variables are reversed) and then each dot will
represent one subject. The scatter plot in Figure 8.7 represents this data. Notice how the line
drawn through the data points has an upward slope. This slope represents the direction of the
relationship and tells us that as experience increases so does income.
Figure 8.7: Scatter Plot for Sample Data
Correlation and Causality. One common mistake made by people interpreting a correlational
coefficient refers to causality. When we see that depression and low self-esteem are negatively
correlated, we often surmise that depression must therefore cause the decrease in self-esteem.
When contemplating this, consider the following correlations that have been found in research:



Positive correlation between ice cream consumption and drownings
Positive correlation between ice cream consumption and murder
Positive correlation between ice cream consumption and boating accidents

Positive correlation between ice cream consumption and shark attacks
If we were to assume that every correlation represents a causal relationship then ice cream
would most certainly be banned due to the devastating effects it has on society. Does ice-cream
consumption cause people to drown? Does ice cream lead to murder? The truth is that often
two variables are related only because of a third variable that is not accounted for within the
statistic. In this case, the weather is this third variable because as the weather gets warmer,
people tend to consume more ice cream. Warmer weather also results in an increase in
swimming and boating and therefore increased drownings, boating accidents, and shark
attacks.
So looking back at the positive correlation between depression and self-esteem, it could be that
depression causes self-esteem to go down, or that low self-esteem results in depression, or that
a third variable causes the change in both. When looking at a correlational coefficient, be sure
to recognize that the variables may be related but that it in no way implies that the change in one
causes the change in the other.
Specific Correlations. Up to this point we have been discussing a specific correlation known
as the Pearson Product Moment Correlation (or Pearson’s r) which is abbreviated with the letter
‘r.’ Pearson is the most commonly cited correlation but can only be used when there are only
two variables that both move in a continuous linear (straight line) direction. When there are
more than two variables, when the variables are dichotomous (true/false or yes/no) or rank
ordered, or when the variables have a nonlinear or curved direction, different types of
correlations would be used.
The Biserial and Point Biserial Correlations are used when one variable is dichotomous and the
other is continuous such as gender and income. The phi or tetrachoric correlations are used
when both variables are dichotomous such as gender and race. And finally, Spearman’s rho
correlation is used with two rank ordered variables and eta is used when the variables are
nonlinear.
Chapter Conclusion
While this chapter only provided a quick basic summary of descriptive statistics, it should give you
a good idea of how data is summarized. Remember, the goal of descriptive statistics is to
describe, in numerical format, what is currently happening within a known population. We use
measures such as the mean, median, and mode to describe the center of a distribution. We use
standard deviation, range, or interquartile range to describe the variability of a distribution, and we
use correlations to describe relationships among two or more distributions.
By knowing this information and by understanding the basics of charting and graphing in
descriptive statistics, inferential statistics become easier to understand. In fact, much of the
information in this chapter becomes the foundation for the advanced statistics discussed in the
chapter 9.