Measures of Variability

1
Measures of Variability

• Chapter 5

2
Objectives

• Be able to define and describe the term
  variability and the measures of variability
• Compute each of the 5 measures of variability
  that we will cover
• Note
• Use the formulas from lecture and the course
  packet, NOT the book.

3
Variability Dispersion

• The degree to which individual data points are
  distributed (spread out) around the mean
• Variability is how spread out the points are

0
50
100
0
50
100
4
Variability Dispersion (cont.)

• Numeric Example
• 70 70 75 80 80 Mean is 75
• 30 80 85 90 90 Mean is 75
• We can see the data are more spread out in the
  second case

5
Measures of Variability

• We will talk about five measures
• Range
• InterQuartile Range
• Average Deviation
• Variance
• Standard Deviation
• We can have sample and population variability
  measures

6
Range

• The distance from the lowest to the highest score
• Range Highest score - Lowest Score
• Same as what we do to determine the number of
  intervals in a grouped frequency table
• A problem with the range is that it is not a very
  precise measure
• It only tells you the spread between the highest
  and lowest score
• Compute the range in the same way for the sample
  population

7
Example

• Because of oil shortages, gas prices in the
  Midwest soar to a high of 5.50 a gallon. Luckily
  gas becomes cheap again and hits a low of 1.00.
• What is the range?
• Range

8
InterQuartile Range (IQR)

• The range of the middle 50 of the observations
  25th to 75th percentile

50
75
25
0
100
This the IQR
9
How to compute the IQR

• First, compute the median for the entire
  distribution
• Second, use the median to split the distribution
  into 2 halves (the scores above the median the
  scores below the median)
• Third, compute the median for each half of the
  distribution. These will be the 25 and 75

10
Example

• Using the following numbers compute the IQR
• 22, 34, 35, 36, 38, 40, 43, 45, 54, 59, 60
• Median (n1)/2
• Lower Set
• Upper Set
• Lower Median (25th percentile)
• Upper Median (75th percentile)

11
IQR (cont.)

• A major problem with the IQR is that it
  eliminates 50 of the scores
• Because it uses only the middle 50, we are
  discarding 25 at each end

50
We Ignore the Top 25
75
25
We Ignore the Bottom 25
0
100
Middle 50
12
Average Deviation

• One intuitive way to determine the spread around
  the mean is see how far each score is from the
  mean
• So this would be (X-X) for each score
• We could then sum each of scores minus the mean
• ?(X-mean)
• DO NOT DO THIS The average deviation will always
  get you zero

13
Example

• 1 2 3 4 5
• Mean ?X/n 15/5 3
• Subtract the mean from each score
• (1-3), (2-3), (3-3), (4-3), (5-3)
• -2 -1 0 1 2
• ?(X-mean) 0

14
Review

• Range Highest score - Lowest score
• InterQuartile Range (middle 50 of the scores)
• Find the median of the distribution use the
  median to split the distribution into 2 halves
• Find the median of each half
• Average Deviation
• Subtract the mean from each score and sum of
  these numbers ?(X-X)
• Dont do this! It will always equal zero
• One way around this is to take the absolute value
  of the average deviation ? X-X

15
Variance

• Sum of the squared deviations about the mean
  divided by sample size-1
• Sample variance - s2
• Definitional formula
• s2 ?(X-X)2
  
• n-1
• Computational formula
• (?X)2
  
• s2 ?X2 - n .
  
• n-1

16
Variance (cont.)

• Population variance (?2)
• Definitional formula
• ?2 ?(X-?)2
• N
• Computational Formula
• (?X)2
  
  ?2 ?X2 - N .
  
• N

17
Standard Deviation (SD)

• It is difficult to interpret the variance because
  it is in squared units
• So , we use the square root of the variance
• Sample standard deviation
• s sqrt(s2)
• Population standard deviation
• ? sqrt(?2)

18
What does the SD tell us?

• The SD tells us how closely scores fall to the
  mean in standardized units
• When we have a fairly symmetric distribution, we
  can make a few generalization about the SD
• We can say that about 66 of the scores will fall
  within 1 SD of the mean
• About 95 within 2 SD of the mean
• About 99.7 within 3 SD of the mean

19
Example
X74.36 s5.12
Mean
1SD
1SD
2SD
2SD
3SD
3SD
79.48
74.36
84.60
89.72
69.24
64.12
50.00
66, 68, 70, 71, 74, 75, 76, 77, 79, 80, 82
20
Extreme Scores, variance, SD

• We saw that extreme scores can pull the mean to
  the extreme scores
• Extreme scores add variability to a distribution
• So extreme scores increase the variance SD

21
Variance, SD, Constants

• If you add, subtract, multiply, or divide a
  constant by every score, it is the same as
  adding, subtracting, multiplying or dividing the
  mean by the constant
• When you add or subtract a constant from each
  score, it has NO effect on the variance or SD
• Why? Variance SD are measures of variability.
  Adding a constant to each score doesnt change
  the overall variability of a distribution

22
Variance, SD, Constants (cont.)

• If you multiply or divide a constant by every
  score, it has the same effect on the SD the
  other measures of variability but not the
  variance
• If you multiply or divide a constant by every
  score, it has the effect of multiplying or
  dividing the variance by the square of the
  constant