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UNIT IV STUDY GUIDE
Data Analysis: Descriptive Statistics
Course Learning Outcomes for Unit IV
Upon completion of this unit, students should be able to:
6. Differentiate between various research-based tools commonly used in businesses.
6.1 Describe various forms of descriptive statistics, including frequency distribution tables,
histograms, descriptive statistics tables, Kolmogorov-Smirnov tests, measurement scales, and
measures of central tendency.
7. Test data for a business research project.
7.1 Establish whether assumptions are met to use parametric statistical procedures by applying
descriptive statistics.
Course/Unit
Learning Outcomes
6.1
7.1
Learning Activity
Unit Lesson
Video: Kolmogorov-Smirnov Test of Normality in Excel
Video: Parametric and Nonparametric Statistical Tests
Video: Checking that Data Is Normally Distributed Using Excel
Video: 3. Choosing Between Parametric & Non-Parametric Tests
Article: “Difference Between Parametric and Nonparametric”
Article: “Deciphering the Dilemma of Parametric and Nonparametric Tests”
Unit IV Scholarly Activity
Unit Lesson
Unit IV Scholarly Activity
Reading Assignment
In order to access the following resources, click the links below:
Fields, H. (2018). Difference between parametric and nonparametric. Retrieved from
Dominguez, V. (2016, April 16). Make a histogram using Excel’s histogram tool in the Data Analysis ToolPak
[Video file]. Retrieved from https://www.youtube.com/watch?v=xekiDJzajYk
Click here for a transcript of the video.
Grande, T. (2017, August 19). Kolmogorov-Smirnov test of normality in Excel [Video file]. Retrieved from
Click here for a transcript of the video.
Grande, T. (2015, July 30). Parametric and nonparametric statistical tests [Video file]. Retrieved from
Click here for a transcript of the video.
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Macarty, M. (2015, September 21). Get descriptive statistics in Excel with Data
Analysis
Toolpak
[Video file].
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Retrieved from https://www.youtube.com/watch?v=h-RzBhBzJOQ
Title
Click here for a transcript of the video.
Oxford Academic (Oxford University Press). (2016, November 17). Checking that data is normally distributed
using Excel [Video file]. Retrieved from https://www.youtube.com/watch?v=EG8AF2B_dps
Click here for a transcript of the video.
Rana, R., Singhal, R., & Dua, P. (2016). Deciphering the dilemma of parametric and nonparametric tests.
Journal of the Practice of Cardiovascular Sciences, 2(2), 95. Retrieved from
http://link.galegroup.com.libraryresources.columbiasouthern.edu/apps/doc/A488649197/AONE?u=ora
n95108&sid=AONE&xid=c54eaf34
The Roslin Institute – Training. (2016, May 9). 3. Choosing between parametric & non-parametric tests [Video
file]. Retrieved from https://www.youtube.com/watch?v=_1mH6CnXKfM
Click here for a transcript of the video.
Unit Lesson
Data Analysis: Descriptive Statistics
The course is now entering the data analysis stage of research design. This is where the methodological fork
in the road goes decisively down the quantitative path. The first topic of discussion under data analysis will be
what is referred to as descriptive statistics. As the name suggests, the researcher describes the data that are
collected. During this stage, the data are described both visually and statistically. Data may be visually
displayed to reveal distribution of data, trends, anomalies, outliers, etc. Visual displays of data may take the
form of graphs, histograms, tables, plots, and other diagrams. This stage is done before any statistical
procedures are used to test the research hypotheses. This begs the question of why the researcher should
not simply jump in and immediately start testing their hypotheses using statistical analysis. The following
explains the importance of descriptive statistics to test data to ensure assumptions are met before using a
parametric test.
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Assumptions: The Importance of Describing Data
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There are various benefits of describing the data. One of the most important benefits is to determine if the
data meet the assumptions that are required for the use of parametric statistical procedures. Parametric
procedures include, but are not limited to, correlation, regression, t test, and ANOVA. Parametric tests have
different assumptions that must be met depending on which test is being considered, but most parametric
tests require that the assumption of normality be met. Normality refers to a normal distribution of data which,
when graphed as
80
frequencies, resembles a
bell shape (as in the image
to the right). Other common
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assumptions that must be
met, depending on the
Bell
statistical procedure used,
60
Curve
include sample size, levelsof-measurement,
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homogeneity of variance,
independence, absence of
40
outliers, linearity, etc. (Field,
2005). It is critical that the
researcher understands the
30
assumptions for any
parametric statistical
20
procedure being considered
to determine if they are met
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before employing the
procedure in a research
study. An Internet search
10 20
30 40 50 60 70 80 90 100
for any parametric test will
quickly return results that
Normal distribution graph with a bell curve
list required assumptions.
If the assumptions are not
met, parametric statistical procedures cannot be used. To use them would result in invalid results.
Fortunately, there are corresponding non-parametric tests that can be used when the data do not meet
assumptions for parametric tests. Non-parametric tests also have assumptions that must be met, but they are
fewer and less rigid. An example of a parametric procedure for correlation would be Pearson’s correlation
coefficient (Pearson’s r), while a corresponding non-parametric test for correlation would be Spearman’s rank
correlation coefficient (Spearman’s rho). An example of a causal-comparative parametric procedure would be
ANOVA, while a corresponding non-parametric causal-comparative test would be Kruskal-Wallis.
Since non-parametric tests do not require that as many assumptions are met, some students wonder why
non-parametric tests are not always used. The reason is that parametric tests are superior to and more
powerful than non-parametric tests and should be used if the assumptions are met. A parametric test is more
likely to find a true effect when one exists, therefore rejecting the null hypothesis, than a non-parametric test
(Norusis, 2008). In other words, a parametric test is less likely to commit a Type II error. Norusis (2008)
recommends that researchers conduct both parametric and non-parametric tests if they are unsure as to
which is most appropriate to use. If the test results are the same, there is nothing more to worry about. If the
test results are statistically significant for the parametric test, and non-significant for the non-parametric test,
the researcher should take a closer look at whether the assumptions were met or not.
Assumption of Normality
Assumptions are evaluated both visually and statistically. As mentioned previously, a normal distribution of
data is the most commonly required assumption for parametric statistical tests. The following will explain how
the assumption of normality can be described and tested.
A normal distribution of data exhibits the characteristics of a bell-shaped curve, as shown below. In a perfect
normal curve, the frequency distribution is symmetrical about the center; the mean, median, and mode are all
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equal; and the tails of the curve approach but do not touch the x-axis (Salkind,UNIT
2009).
These are
all
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preliminary indicators that a curve may represent a normal distribution, but there
are additional factors to
Title
consider.
Distribution curves can be short and wide, tall and thin, and anywhere in between. As shown below, each of
the colored bell-shaped curves has a mean
(μ) of zero. Their standard deviations (σ),
however, or the measure of how widely the
data disperses around the mean, are
different for each curve. The orange curve
has a relatively small standard deviation
because the data is closely clustered around
the mean. The red curve has a relatively
large standard deviation because the data is
loosely clustered around the mean.
Kurtosis describes the tallness of the curves.
A platykurtic curve is short and squatty (think
plateau), which, as shown at the right in the
red curve, represents a relatively greater
number of scores in the tails of the curves. A
leptokurtic curve is tall and thin (think leapt
for the sky), which, as shown in the orange
Distribution curves
curve, represents a data distribution of
relatively fewer number of scores in the tails (Field, 2005). Platykurtic and leptokurtic curves can challenge
the assumption of normality, even when the curve is bell-shaped.
The data may also be asymmetrical with the data more heavily distributed to one side of the curve or the
other. When the data distribution curve is asymmetrical, it is referred to as skewness. Below are examples of
negative skewness and positive skewness. Like platykurtic and leptokurtic curves, those exhibiting skewness
also threaten the assumption of normality.
Left-skewed and right-skewed graphs
(Sundberg, 2014)
The assumption of normality can be evaluated visually by describing the frequency of responses in a data set.
The frequency table below shows the results of a 120-point safety test administered to 500 employees. For
example, two employees scored in the test range of 50–54, 90 employees scored in the range of 85–89, and
three employees scored in the range of 110–114.
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When the frequency data is plotted in a histogram, the curve of the
data can be observed. To create a histogram, the data values (test
score ranges) from the data set are plotted on the x-axis, and the
frequency of the values are plotted on the y-axis. So, using the same
example from the discussion of the frequency table, it can be seen in
the histogram that two employees scored in the test range of 50–54,
90 employees scored in the range of 85–89, and three employees
scored in the range of 110–114.
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By observing the histogram below, it appears the data are
approximately normally distributed, and there are no visible outliers.
While there is no skewness observed, the kurtosis favors a
leptokurtic curve. Skewness and kurtosis can be confirmed by
generating descriptive statistics, which is a routine function in
statistical packages, including Excel Data Analysis Toolpak. There is
a lot of debate regarding acceptable levels of skewness and kurtosis
among researchers. George and Mallery (2010) suggest skewness
and Kurtosis scores between -2 and +2 as satisfactory results to
accept normal distribution. All researchers agree that the closer
skewness and kurtosis are to 0, the better. The more kurtosis and
skewness deviate from 0, the greater the chances that the data is
not normally distributed (Field, 2005). As shown in the descriptive
statistics table, both skewness and kurtosis are both relatively close
to 0.
It should also be noted that the mean, median, and mode are similar in the descriptive data table below. As
noted above, the mean, median, and mode are identical in a perfect distribution. The data presented here
would suggest that it is approximately a normal distribution of data.
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Descriptive Statistics
Mean
Standard Error
Median
Mode
Standard Deviation
Sample Variance
Kurtosis
Skewness
Range
Minimum
Maximum
Sum
Count
Largest(1)
Smallest(1)
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80.546
0.446621439
81
75
9.986758969
99.73535471
0.095314585
0.065078019
64
53
117
40273
500
117
53
The frequency distribution should also be observed for outliers. Outliers are extreme scores far away from the
mean in the left or right tails of the curve. Outliers can bias the mean due to their extreme scores. There are
different recommendations for how to treat outliers, such as removing the outlier from the data set, but the
ramifications should be understood before taking any such action. This is an example where consulting the
literature is strongly recommended.
Finally, normality can be tested statistically. Several tests can be used to objectively test for normality
including Kolmogorov-Smirnov, Shapiro-Wilk, chi-square, Jarque-Bera, Anderson-Darling, and others. Each
test has advantages and disadvantages. Once again, this is where the researcher is well-served to consult
the literature to determine the most appropriate test for his or her project.
The Kolmogorov-Smirnov (KS) test is often used to test for normality. KS compares the frequency distribution
of the sample data set to a model of normally distributed data with the same mean and distribution as the
sample data. The KS test is performed to test a null and alternative hypothesis, like any other statistical test.
The following are the hypotheses.
Ho1: There is no statistically significant difference in normality between the sample data and model data.
Ha1: There is a statistically significant difference in normality between the sample data and model data.
If the results are statistically significant at a p level < .05, the null hypothesis is rejected, and the alternative
hypothesis is accepted that there is a statistically significant difference in normality between the sample data
and model data. Therefore, we would conclude that the assumption of normality is not met, and a nonparametric test would be required to test our data.
If the results are not statistically significant at a p level > .05, the null hypothesis is accepted (and the
alternative rejected) that there is no statistically significant difference in normality between the sample data
and model data. Therefore, we would conclude that the assumption of normality is met, and a parametric test
would be acceptable to test our data.
It is important to note that the above steps for evaluating the assumption of normality require a holistic view.
No single description of the data is sufficient to make a decision about normality. For example, the KS test is
sensitive to small changes in normality for large sample sizes. The result is that it can be prone to Type I
errors. Therefore, the researcher should consider all the available information, both visual inspection and
statistical analysis, before making a decision about normality (Field, 2005). If, after following the steps above,
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the assumption of normality does not appear to be met, non-parametric statistical
should be
UNITprocedures
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considered in lieu of parametric tests.
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Assumptions Other Than Normality
There are two additional assumptions that should be met for any statistical test. They are measurement
scales and measures of central tendency.
Measurement scales: Statistical procedures used to test hypotheses have unique assumptions about the
scales on which the data are measured. Data are measured on nominal, ordinal, interval, or ratio scales. It is
important to determine the assumption of measurement scales for any statistical procedure being considered
to test the data. For example, an assumption of Pearson’s r is that data be measured at the interval or ratio
level. Pearson’s r could not be used to analyze ordinal data. The non-parametric test, Spearman’s rho, would
be required to analyze ordinal data for correlation.
Rules for Measurement Scales
Nominal: Nominal data can be classified but not ordered and have no meaningful distance between variables
or unique origin (true zero). This is also referred to as categorical data. Examples include names or
categories, like gender and marital status. Examples of statistical procedures that use nominal data include
chi-square (Cooper & Schindler, 2014).
Ordinal: Ordinal data can be classified and ordered but have no meaningful distance between data values or
unique origin (true zero). Examples include surveys with responses ranked on a five-point Likert scale, such
as strongly agree to strongly disagree. Examples of statistical procedures that use ordinal data include
Spearman’s rho, Mann-Whitney test, Wilcoxon test, Kruskal-Wallis test, and Friedman test (Cooper &
Schindler, 2014).
Interval: Interval data can be classified and ordered and have meaningful distance between data values but
no unique origin (true zero). A classic example of an interval level of measurement is temperature measured
in degrees. The data is ordered, there are differences between measures, but there is no true zero. Since
there is no true zero, it would be improper to say 40 degrees is twice as cold as 20 degrees. Examples of
statistical procedures that use interval data include Pearson’s r, regression analysis, t test, and ANOVA
(Cooper & Schindler, 2014).
Ratio: Ratio data can be classified and ordered, have meaningful distance between data values, and have
unique origin (true zero). Examples include age in years and income in dollars. Examples of statistical
procedures that use ratio data include Pearson’s r, regression analysis, t test, and ANOVA (Cooper &
Schindler, 2014). It should be noted that parametric tests are used to analyze data measure at the interval
and ratio levels but cannot be used to analyze data measured at the nominal and ordinal levels.
Measures of central tendency: It may have become evident by now, from the use of the histogram and the
discussion of normality, that there is interest in how the data points are dispersed around the mid-point of the
curve. This is called central tendency and is the foundation for statistical analysis using linear models. In
short, our statistical procedures evaluate how much our data vary from that midpoint when a straight line is fit
to the data (Field, 2005). The important takeaway is that the central tendency of that midpoint can be
measured in three different ways: a) mean, b) median, and c) mode. As was seen in the descriptive statistics
output above, mean, median, and mode are usually included in descriptive statistics generated by software.
As was the case with normality and levels of measurement, it is important to determine the assumption of
central tendency for any statistical procedure being considered to test the data.
Mean: The arithmetic mean is the most commonly used measure of central tendency. It is calculated by
adding the data scores and dividing by the number of cases. The mean is the measure of central tendency
used with interval and ratio data and is used for statistical procedures like correlation, regression analysis, t
test, and ANOVA (Salkind, 2009).
Median: The median is the score among the distribution of data, when ordered from highest to lowest, where
half of the data points occur above the median and half of the data points occur below the median. In the data
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set 1, 3, 5, 7, and 9, the median would be 5 since half of the values occur above
andx half
below.
The median
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is the measure of central tendency used with ordinal data (Salkind, 2009).
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Mode: The mode is the data value that occurs most frequently in the data set, regardless of order. In a data
set of 5, 5, 5, 3, 3, 9, 9, 9, 9, 1, 1, 1, 7, 7, 7, 7, 7, the mode would be 7 because it is the value that occurs
most frequently in the data set. The mode is the measure of central tendency used with nominal levels of
measurement (Salkind, 2009).
In Closing—A Word About Validity and Reliability
Although some of the most important and common assumptions of statistical testing have been discussed in
this lesson, there are still more. This may seem like a very taxing and laborious process to partake in before
even getting to the point of testing the research hypothe …
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