Assignment: Measures of Variability
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Calculating Measures of Variability
Your Name
Bachelor of Science in Criminal Justice, Walden University
CRJS-3004: Data Analysis for Criminal Justice Professionals
John Walker
May 09, 2021
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Calculating Measures of Variability
Sample
Person Number of
Arrests
Bo 0
Johnny 1
Sarah 3
Julie 5
Barbar a
0
Ken 2
Keith 8
Nicole 2
Sam 1
Stuart 3
CALCULATIONS
[Sample size: 10]
Range: 8
Lowest value: 0 Highest value: 8
Interquartile range: 3.25
Lower quartile (Q ) or 25%: 1.25₁ Middle quartile (Q ) or Median: 2₂ Higher quartile (Q ): or 75%: 4.5 ₃
Variance (s²): 6.0555556
Standard deviation (s): 2.46
Mean: 0+0+1+1+2+2+3+3+5+8 = 25 and
25/10 = 2.5
Median or (Q ): [0,0,1,1,(2,2),3,3,5,8] = 2₂
Modes: [0, 1, 2, 3] (there are two 0’s, two one’s, two 2’s, and two three’s). They are the numbers in the data set seen more frequently.
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Calculating Measures of Variability
A summary statistic that describes the amount of dispersion in a dataset is known as a
measure of variability. It asks what is the degree of dispersion of the values? The term
"variability" refers to how dispersed or spread out a set of data is. Measures of variability
determine how far away the data points appear to fall from the center, while measures of central
tendency identify the normal value (Bachman & Schutt, 2019). To calculate measures of
variability you need to find the range, interquartile range, variance, and standard deviation of the
samples you are researching.
The range is a measure of dispersion, or how often the values in a data set are likely to
deviate from the mean. It is the difference between the largest and smallest values in a dataset.
By subtracting the lowest from the highest value in the package, the range can be easily
determined (Bachman & Schutt, 2019). For example:
Range = maximum (xh) – minimum (xl)
(x) represents the value of the number in the data set
The range for the number of arrests would be [8(max) – 0(min)] = 8.
Answer: Range = 8
When a dataset contains a value that is much higher or lower than the rest of the dataset this is
called an outlier. The best measure of variability to use when this occurs is the interquartile
range (IQR). The IQR is the range for the middle or 50 percentiles of the data. It will only
consider the middle values therefore it is not influenced by the outliers (Bachman & Schutt,
2019). To calculate for this sample, it would look like this: IQR = (Q - Q ), Q being the higher ₃ ₁ ₃
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quartile or 75 percentile and Q being the lower quartile or 25 percentiles. We know the median ₁
equals two because it is the middle or center of the data set. For example, IQR = (Q - Q ): ₃ ₁
Put the sample numbers in order from least to greatest; [0, 0, 1, 1, (2, 2), 3, 3, 5, 8].
Lowest quartile (Q ) or 25% of values in the data set smaller than Q : [0, (0, 1), 1,] = ₁ ₁
1.25
Highest quartile (Q ) or 75% of the observations less than Q : [3, (3, 5), 8] = 4.5₃ ₃
IQR = (4.5 – 1.25)
Answer: IQR = 3.25
Variance measures dispersion of data from the mean. The formula for the variance is sum of
squared deviations from the mean divided by the size of the data set.
s² or the variance of sample: s² = ∑(xi – ¯x)²/(n-1)
∑ or the sum of the squared values (x’s)
¯x or mean
n: the number of people in a sample (Statistical Concepts,
2021).
For example, s² = ∑(xi – ¯x)²/(n-1):
¯x = ∑(xi)/n: add the sum of the data and divide by the amount of data.
[0+0+1+1+2+2+3+3+5+8] = 25/10 = 2.5
¯x = 2.5
xi - ¯x: subtract the mean (2.5) from the number of arrests:
0-ˉ2.5, 0-ˉ2.5, 1-ˉ1.5, 1-ˉ1.5, 2-ˉ0.5, 2-ˉ0.5, 3-0.5, 3-0.5, 5-2.5, 8-5.5
DATA xi - ¯x (xi - ¯x)²
0 -10 100 0 -10 100 1 -9 81 1 -9 81 2 -8 64 2 -8 64 3 -7 49 3 -7 49 5 -5 25 8 -2 4
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(xi - ¯x)²: then square each answer
-2.5² = 6.25, -1.5² = 2.25, ˉ0.5² = .25, 0.5² = .25 2.5² = 6.25, -5.5²
to get the ∑ (sum) you add the (xi - ¯x)² column:
6.25+6.25+2.25+2.25+.25+.25+.25+.25+6.25+30.25 = 54.5
s² = 54.5/9
Answer: Variance = 6.0555556
Standard deviation is the average of all the deviations for the mean. It is the summary of the
most common variation of the entire sample or the square root of the variance. When the
distribution is normal or close to normal, the standard deviation is hugely helpful as a measure of
variability since the proportion of the distribution within a specified number of standard
deviations from the mean can be measured (Lane, n.d.). A large Standard deviation suggests that
the mean is a weaker measure, and a small standard deviation suggests the mean is a stronger
measure. For instance:
Bo Johnny Sarah Julie Barbara Ken Keith Nicole Sam Stuart 0
1
2
3
4
5
6
7
8
9
'Times Arrested' by 'Sample'
Sample
T im
e s
A rr
e st
e d
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The distance between Bo and Keith is large meaning that the mean or average is a weaker
measure. Where the distance between Bo and Barbara remains the same making the mean the
stronger measure.
The formula for standard deviation is:
s = ∑(xi−¯x)²/(n−1)
s = s²/10−1
s²= 54.5/9
s = √6.0555556
Answer: Standard Deviation (s) = 2.4608038
We know the difference between maximum and minimum in distribution is the range. The
average distance of scores in distribution from their mean is standard deviation and the square of
the standard deviation is variance. Once calculations are made then we need to determine what
the distribution and variance of this data are. The sample contains ten different people and how
many times they have been arrested. The data shows us that on average out of these ten people,
the average person has been arrested 2.5 times. Finding the interquartile range for the large
distance in the range and figuring out the standard deviation using the datasets variance, plus the
mean all show that the average sum of arrests is 2.5.
Knowing how to calculate this data can help me in many ways in the juvenile justice field.
Being able to figure out on average how many juvenile delinquents are male or female, what part
of town has more juvenile delinquents, or does mental illness contribute to delinquency. The list
can go on and on with how knowing how to calculate the measures of variance and measures of
tendency can help contribute to finding solutions for problems in the criminal justice system. In
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addition to when I better understand the data being presented, I can then distribute the data to the
public in a statistical way, such as, like with the sample. It is much easier to tell people that on
average this certain group was arrested two and a half times than it is to just list the dataset.
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References
Bachman, R. D., & Schutt, R. K. (2019). The practice of research in criminology and criminal
justice (7th ed.). (Ch. 14). (pp. 416-425). Thousand Oaks CA: SAGE Publications.
Lane, D. (n.d.). Measures of variability. Retrieved May 04, 2021, from
http://cnx.org/content/m10947/latest/
Statistical Concepts. (2021). Statistics formulas. In Statistics. Retrieved from
http://statisticalconcepts.blogspot.com/2010/12/statistics-formulas.html
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