Computing Descriptive Statistics © 2014 Argos.docx

Computing Descriptive Statistics
© 2014 Argosy University
Page 2 of 5
Research and Evaluation Design
©2014 Argosy University
2 Computing Descriptive Statistics
Computing Descriptive Statistics: “Ever Wonder What Secrets
They Hold?” The Mean, Mode, Median, Variability, and
Standard Deviation
Introduction
Before gaining an appreciation for the value of descriptive
statistics in behavioral science
environments, one must first become familiar with the type of
measurement data these statistical
processes use. Knowing the types of measurement data will aid

the decision maker in making
sure that the chosen statistical method will, indeed, produce the
results needed and expected.
Using the wrong type of measurement data with a selected
statistic tool will result in erroneous
results, errors, and ineffective decision making.
Measurement, or numerical, data is divided into four types:
nominal, ordinal, interval, and ratio.
The businessperson, because of administering questionnaires,
taking polls, conducting surveys,
administering tests, and counting events, products, and a host of
other numerical data
instrumentations, garners all the numerical values associated
with these four types.
Nominal Data
Nominal data is the simplest of all four forms of numerical data.
The mathematical values are
assigned to that which is being assessed simply by arbitrarily
assigning numerical values to a
characteristic, event, occasion, or phenomenon. For example, a
human resources (HR) manager
wishes to determine the differences in leadership styles between
managers who are at different
geographical regions. To compute the differences, the HR
manager might assign the following
values: 1 = West, 2 = Midwest, 3 = North, and so on. The
numerical values are not descriptive of
anything other than the location and are not indicative of
quantity.
Ordinal Data
In terms of ordinal data, the variables contained within the

measurement instrument are ranked in
order of importance. For example, a product-marketing
specialist might be interested in how a
consumer group would respond to a new product. To garner the
information, the questionnaire
administered to a group of consumers would include questions
scaled as follows: 1 = Not Likely, 2
= Somewhat Likely, 3 = Likely, 4 = More Than Likely, and 5 =
Most Likely. This creates a scale
rank order from Not Likely to Most Likely with respect to
acceptance of the new consumer
product.
Interval Data
Oftentimes, in addition to being ordered, the differences (or
intervals) between two adjacent
measurement values on a measurement scale are identical. For
example, the differences in age
between managers 25 years of age and 30 years of age are the
same as the differences in age
between managers who are 40 years of age and 45 years of age.
That is to say, when each
interval represents the same increment of that which is being
measured, the measure is referred
to as an interval measurement or interval mathematical value.
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Ratio Data
Some measurements, in addition to having an equal interval
value, also have an absolute zero
value. In this instance, zero represents the absence of the
variable being measured. With an
absolute zero value, for example, either you have some money
or you do not have any money. In
the money scenario, adding the interval quality to ratio data
would mean that you have no money,
$1.00 to $5.00, $6.00 to $10.00, and so on. What is being stated
here is that ratio data is
quantitative as it tells us the amount of the variable being
measured. Consider other examples of
ration data such as the percentage of votes received by a
candidate, the gross national product
per capita, the current American crime rate, and the number of
finished consumer products
manufactured per day per person—all of these are examples of
ratio data.
Why Measurement Data Matters
To the behavioral scientist, when conducting research, the level
of measurement is important
because the higher the level of measurement of a variable, the
more powerful the statistical
techniques that can be employed to analyze the data. Take, for
example, the voters’ choice
wherein the nominal variables in the 2004 presidential race
were Bush, Kerry, Nader, etc. One
can count the number of votes each candidate received as well
as calculate the percentage each

candidate received. One can also calculate joint frequencies and
percentages by region and by
gender. One can also calculate the relationship between region
and vote and whether the
relationship occurred by chance. Unfortunately, using nominal
measurement data does not permit
one to use advanced methods of statistics. In the example
presented above, even when we
assign numbers to each candidate, we cannot very well
determine that Bush plus Kerry equals
Nader or that Nader divided by Kerry is half way between Bush
and Kerry, and so on. In an
attempt to calculate the situation, only the mentioned higher
level statistical techniques are
required.
If one uses a technique that assumes a higher level of
measurement than is appropriate for the
data, there is a risk of getting meaningless results and answers.
At the same time, if one uses a
technique that fails to take advantage of a higher level of
measurement, important things are often
overlooked about the data collected.
Computing and Using the Mean, Median, Mode, Frequency
Distribution, and Standard
Deviation
Regardless of the type of measurement data garnered from a
data collection set, the mean,
mode, median, frequency distribution, and standard deviation
can be calculated. However, simply
because mathematical calculations can be made does not imply
the legitimacy of their use. The
remainder of this section will deal with how one computes and
uses measures of central tendency

in business situations.
Mean
Mean is the arithmetic average of a group of measurement
values. To be meaningful, the
resulting mathematical value must be based on at least an
ordinal set of data or above. Interval
and ratio are also appropriate for calculating the mean. The
resulting mean of a group of data will
only describe the data in general descriptive terms. The mean
alone cannot be used to draw
conclusions and make inferences about a population being
studied.
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Formula:
N (n) = Number of participants in a study, or numeric values
Σ = Sum of all numeric values
X = Raw score, or individual measurement score

= The mean
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Example
Consider a production manager who wants to determine whether
the 11 employees of the second
shift of line employees are producing more baby strollers than
the 11 first-shift employees. The
second-shift employees produce the following number of baby
strollers: 5, 7, 8, 5, 7, 7, 9, 9, 5, 5,
5. The first-shift employees produce the following number of
units: 7, 7, 7, 7, 6, 6, 9, 2, 4, 6, 8.
Solution
:
Second shift total units (ΣΧ) = 72

Second shift total number of employees (n) = 11
Mean = 72/11 = 6.55 units produced
First shift units (ΣΧ) = 69
First shift total number of employees (n) = 11
Mean = 69/11 = 6.27 units produced
Conclusion
What appears to be happening is that the second shift of line
employees produces more baby
strollers than the first shift of line employees.
Caution: No other conclusions can be drawn from the mean. To
determine whether production
differences are significant, a higher level statistical process
must be used. We can only “describe”
what has happened here and cannot draw conclusions or make
inferences as to why or how
much.

Median
Median is broadly defined as the middle value of a set of
measurement values. Just like medians
divide roads down the middle, so does the median in statistics
in that the median is simply the
middle number For highly skewed distributions, the median is a
better measure of central
tendency than the mean as extreme outliners or measurement
values do not affect it.
Formula:
No statistical formula is needed. To calculate the median of a
group of measurement scores,
simply find the
midpoint of the distribution by arranging the scores in
ascending order—from low to high.
Example
Second-shift stroller production 5, 7, 8, 5, 7, 7, 9, 9, 5, 5, 5
Ordered Values –5, 5, 5, 5, 5, 7, 7,
7, 8, 9, 9

Median = 7
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First-shift stroller production 7, 7, 7, 7, 6, 6, 9, 2, 4, 6, 8
Ordered values –2, 4, 6, 6, 6, 7, 7, 7, 7, 8, 9
Median = 7
Note: As the median values are equal, the mean is a better
choice to describe the measurement
data.
Mode

Mode is defined as the most frequent value in a measurement
data set and is of limited value.
Example
Second-shift stroller production: 5, 7, 8, 5, 7, 7, 9, 9, 5, 5, 5
Mode = 5
First-shift stroller production: 7, 7, 7, 7, 6, 6, 9, 2, 4, 6, 8
Mode = 7
Variability
Furthermore, the central tendency is a summary measure of the
overall quantity of a
measurement data set. Variability (or dispersion) measures the
amount of spread in a
measurement data set. Variability is generally measured using
three criteria: range, variance, and
standard deviation.
Example

Range
The difference between the largest and the smallest value in the
data set is calculated by
subtracting the smallest value from the largest measurement
value. Although the range is a crude
measure of variability, it is easy to calculate and useful as an
outline description of a data set.
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Example

Second-shift stroller production: 5, 7, 8, 5, 7, 7, 9, 9, 5, 5, 5
Range = 9 – 5 = 4
First-shift stroller production: 7, 7, 7, 7, 6, 6, 9, 2, 4, 6, 8
Range = 9 – 2 = 7
Variance
Variance is a deviation. It is a measure of by how much each
point frequency distribution lies
above or below the mean for the entire data set:
Note: If you add all the deviation scores for a measurement data
set together, you will
automatically get the mean for that data set.
In order to define the amount of deviation of a data set from the
mean, calculate the mean of all
the deviation scores, i.e., the variance.
Formula:

Standard Deviation
In statistics, the standard deviation represents the measure of
the spread of a set of
measurement values from the mean of the data set. Putting it
another way, the standard deviation
can be defined as the average amount by which scores in a
distribution differ from the mean while
ignoring the sign of the difference, i.e., the plus or minus value.
Further, standard deviations are
only good when referring to single data or measurement values,
i.e., finding out when a single
score falls with reference to being above or below the mean.
To find out the standard deviation of a data set, you must
perform the following steps:
1. Calculate the mean of all the scores.
2. Find the deviation of each score from the mean.
3. Square each deviation.
4. Calculate the average of each deviation.
5. Calculate the square root of the average deviation.

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Formula:
Example
Consider, for example, a real estate manager who wants to
determine where his or her
department employees are placed with respect to the average
number of real estate closings they

accomplish per month.
The monthly closings are as follows: 8, 25, 7, 5, 8, 3, 10, 12, 9
1. First, calculate the mean and determine N.
2. Remember, the mean is the sum of scores divided by N,
where N is the number of scores.
3. Therefore, the mean = (8+25+7+5+8+3+10+12+9) / 9 or 9.67
4. Then, calculate the standard deviation, n, as illustrated
below.
Squared
Score Mean Deviation* Deviation
8 9.67 –1.67 2.79
25 9.67 +15.33 235.01
7 9.67 –2.67 7.13
5 9.67 –4.67 21.81
8 9.67 –1.67 2.79

3 9.67 –6.67 44.49
10 9.67 +.33 .11
12 9.67 +2.33 5.43
9 9.67 –.67 .45
(*deviation from mean = score – mean)
Sum of squared deviation = 320.01
Standard Deviation = Square root (sum of squared deviations /
(N – 1))
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= Square root (320.01/ (9 – 1))
= Square root (40)
= 6.32
Conclusion
Real estate closings for the month vary ±6.32 closings above or
below the mean of a 9.67 closing
average. Caution must be exercised here as no conclusion can
be drawn as to whether this is an
acceptable range of real estate closing activity. In other words,
you cannot draw a conclusion
about whether the closings are profitable, not profitable, or
represent an industry average. Further
statistical data analysis would have to be conducted to draw any
such conclusion.
Page 10 of 5

The Color of Data: Visually Reported Data for Descriptive
Statistical Presentations
The Pros and Cons of Visually Reported Data
For the behavioral scientist, whether in business, forensics,
sociology, or anthropology, a host of
other related facts such as profit reporting, result forecasting,
time sequences, overtime hours,
personality test scores, anxiety scores, and IQ are often reported
visually. The reason for using
visual presentations is not only because “a picture is worth a
thousand words” but also because
visually presented data is not generally bogged down with
useless facts and figures. That is to
say, visual presentations usually present the most salient or
robust features of an event,
occurrence, phenomenon, situation, problem, or condition.
Visually presented information can

also be considered the road map to what has happened or lies
ahead. Further, visually reported
data are not only colorful but also easy to construct, and one
does not have to be a statistician to
create them.
Unfortunately, visually reported data results in the behavioral
science arena have two primary
drawbacks, namely, the presenter and the lack of data
sophistication. For the most part, visually
presented data are based on percentages, raw numeric data
scores, and frequencies. Rarely is
visually presented material based on true value statistics based
on inferential statistical findings.
The reason is that the values for inferential statistics are not
amenable to graphs and charts, and
their values speak mathematically for themselves. That is to
say, inferential statistical values are
mathematical values that must be interpreted and are not
subjected to graphic presentation.
A Prelude to Statistical Data Analysis: Raw Data’s Wardrobe in
the Form of Bar Graphs,
Pie Charts, Line Graphs, Stem and Leaf Displays, and Box and
Whisper Plots

Introduction
Regardless of the type of chart or graph you use to illuminate or
express a concept or idea, they
all have one thing in common, namely, to communicate via
picture what is being studied or what
is happening. A behavioral scientist, or any other person who
wants to show what is taking place,
makes use of bar graphs, pie charts, line graphs, leaf displays,
and box and whisper plots.
Unfortunately, however, those who are not well-grounded and
informed about statistical
processes oftentimes rely on these pictorial presentations to
draw conclusions or make inferences
about what is being studied and evaluated. When this happens in
the behavioral science arena,
wrong decisions are often made about important matters.
Nonetheless, graphs and charts are an
important step in achieving what statistical processes will
eventually resolve.
Bar Graph
Bar graphs allow and encourage a great deal of poetic license to
those who design or make use

of them. The reason is that the one designing the bar graph
determines what scale is to be used.
This means that the information can be presented in a
misleading way. For example, by using a
smaller scale (for example, having each half inch of the height
of a bar represent 10 widgets
versus 50 widgets produced), one can exaggerate the truth,
make production differences look
more dramatic, or even exaggerate values. On the other hand, by
using a larger scale (for
example having each half inch of a bar represent 50 widgets
versus 10 widgets produced), a
person can downplay differences, make the end results look less
dramatic than they actually are
or even make small differences appear to be nonexistent. When
evaluating a bar graph, one
should do the following:
Page 11 of 5

Make sure that the bars that divide up values are equal in width
for an equitable comparison.
Make sure that there is an appropriate representation of all
information being presented.
Knowing that the information being presented might not be a
fair representation of all
information, be prepared to dig deeper and use more advanced
statistical processes.
Example
Suppose, for example, we want to pictorially present, via a bar
graph, information on how much
money is spent on transportation by individuals of different
income levels. The first step is to
gather the information from a representative sample (sampling
will be discussed later in the

course) and determine how much money each participant spends
on transportation in a year. The
second step is to define income level. The third step is to
determine the horizontal axis and the
vertical axis. When you are seeking the “how” of something,
always remember that it becomes
the vertical axis and the “category” becomes the horizontal axis.
For this particular example, the
bar graph might possibly look like the following:
To construct a bar graph, go to your Windows task bar, click on
Insert, click on Chart, and enter
your information when prompted.
Pie Charts
Similar to bar graphs, pie charts, or circle graphs, are a pictorial
means using which the individual
can present information in a simple, nonstatistical form.
Further, like bar graphs, pie charts are
usually employed to compare percentages of the same whole.
Unlike bar graphs, pie charts do
not use a set of axes to plot information or data points. Pie

charts are display percentages and
are used to compare different parts of the same whole. With pie
charts, it is important to
remember how they are sometimes misrepresented in business
situations, namely leaving out
parts of the whole and not defining what the whole really is.
When a part of the whole is omitted,
then it increases the percentage values of the remaining parts
that are displayed. When the whole
Page 12 of 5
is not well-defined, the reader is unclear about what the parts
represent. Again, a pie is a
graphical representation of how many individual parts
contribute to the total.

Example
Take, for example, a human resource manager who is interested
in finding out how three different
departments in a business situation waste time on the Internet
on a given day when they should
be engaged in company business. The human resource person
would collect data through a time
study process and determine the number of times each employee
in each department logged on
and off the Internet for personal business. The times would be
collected and added together, and
each department’s time would be converted to percentages.
Going further, the human resource
manager reported that, cumulatively, the employees of
Department 1 spent a total of five hours a
day on the Internet, those of Department 2 spent two hours a
day, and those of Department 3
spent six hours. The pie chart would look like the following:

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