Measures of Variability
A measure of variability is a value that tells us how spread out the scores are in a
distribution. It is a measure of distance. Contrast this to a measure of central
tendency which tells us where the distribution is centered.
A
B
C
0
2
6
10
12
4
5
6
7
8
6
6
6
6
6
If we calculate the sample mean for each sample we see that each sample has a
mean equal to 6.0. However, these distributions look very different in terms of
variability. We can see that in distribution C there is no variability because every
score is the same. There appears to be more variability (spread) in distribution A
compared to distribution B.
We are going to highlight 3 measures of variability.
1.
Range (R)
-the measurement of the width of the entire distribution.
R = highest score – lowest score.
RA = 12
RB = 4
RC = 0
One of the limitations of the range as a measure of variability is that it is based on
the two extreme most scores (highest/lowest) and is very sensitive to outliers: an
extreme score or scores that stand apart from the rest of the distribution. For
example, in distribution B above, if we had a score of 20, the range would be 16.
While this shows the width of the entire distribution it would not tell us that the
rest of the scores are close together and there was an outlier of 20.
2. Interquartile Range (IQR)
-the range of the middle 50% of the distribution. It is the
difference between the score at the 75th percentile and the score at
the 25th percentile.
Here is a method by hand to calculate the interquartile range.
***First you must arrange the scores according to magnitude (low to high or high
to low)
Example: 9 9 8 7 6 5 5 4 3 3 2
1. Find the median. This will mark the midpoint of the distribution. Half of
the distribution falls above this value and half will fall below this value. The
median is not a part of the calculation. It is used to help you divide the
distribution in half.
(9 9 8 7 6) 5 (5 4 3 3 2)
2. Next, find the median of the upper and lower halves.
(9 9 8 7 6) 5 (5 4 3 3 2)
3. Subtract these two scores for the IQR: 8-3 = 5.
The range of the middle 50% of the distribution is 5.
While this IQR eliminates the problem of outliers, the limitation is that we have
left out 50% of the distribution.
If our goal is to find out how spread out the scores are in a distribution, then we
need a technique that would look at every score in the distribution.
3.Variance
-the average of the squared deviations of scores around the mean.
Sample variance
s2 =
̅̅̅̅̅
2
(𝑋−𝑋)
𝑛−1
Definitional formula: defines process
In this formula we are subtracting the mean from every score, squaring those
differences, adding the squared values, and then dividing by n-1.
Let’s take a look at another example.
maze running time/minutes
X
𝑋̅
(X - 𝑋̅)
(X - 𝑋̅)2
9
5
4
16
8
5
3
9
6
5
1
1
4
5
-1
1
2
5
-3
9
1
5
-4
16
__
___
∑X =30
∑(X - 𝑋̅)2 = 52
52
S2= 6−1 =
52/5 = 10.4
Variance measures the distance to the mean in squared units of
measurement. Because we don’t tend to talk about things in squared units we take
the square root to return to the original unit of measurement. When we square
root the variance, we get the sample standard deviation.
S = √𝑠 2
S=√10.4 = 3.22
Much better! On average, the scores are deviating from the mean by 3.22
minutes!
This is a good time to talk about rounding and decimal places: You must take
all of your answers to at least two decimal places. Don’t round to tenths or whole
numbers. If you take your answers to more than two decimal places that is fine.
For rounding: if you had 10.445 or higher then you would round up to 10.45.
If you had 10.444 or lower than you would leave the answer 10.44.
Ok, now that we’ve been through the above example, let’s now calculate the
sample variance with a different formula. This is the formula that we will be
computing from this semester.
Take a look at your formula sheet. See this formula for sample variance?
2
S =
Σ𝑋 2 −
(Σ𝑋)2
𝑛
𝑛−1
Computational formula: formula we use to compute
Let’s work through this one together using the same data from above.
maze running time/minutes
X
9
8
6
4
2
1