Numerical Measures of Variability

Measures of Variability

Measures of Variability are a set of characteristics that examine the dispersion or spread of the distribution that the researcher may be interested in.

This group describes the amount of variability (spread, dispersion, difference) in a set of values.

Range, Interquartile Range, Variance, Standard Deviation, and Coefficient of Skewness

Range

The range is the difference between the highest value and the lowest value.

Ex. If the highest value is 20 and the lowest value is 2, the range is 18.

The range can be reported as 18 or that the “values range from 2 to 20”.

Interquartile RangeThe interquartile range measures the range of the middle 50% of an ordered data set.

not effected by outliers

still preserves the idea of the range

In finding the interquartile range, we actually locate the 25th, 50th, & 75th quartiles

Interquartile Range

Steps to finding the Interquartile Range

1. Rank order the data, Find the Median, Mark that value with Q2 (this is the 50th quartile)

2. Find the Median of the Lower 50% w/o including the Q2 value, Mark that value with Q1 (this is the 25th quartile)

3. Find the Median of the Upper 50% w/o including the Q2 value, Mark that value with Q3 (this is the 75th quartile)

4. IQR = ( Q3 – Q1 )

Variance and Standard Deviation

Most commonly reported and utilized measures of variability

Can be influenced by outliers

Variance and Standard Deviation

If the data is widely scattered,

larger values

Variance

The variance is the average squared amount of deviation from the mean. Generally used as a step toward the calculation of other statistics.

Sample

Variance (s2 ) = (x – x )2 / n - 1

Population

Variance (σ2) = (x - µ)2 / N

x2 = deviation scores squared

Standard Deviation

The standard deviation is the average amount of deviation from the mean. It is the square root of the variance.

Sample

Standard Deviation (s) = (x - x)2 / n - 1

(s) = S2

Population

Standard Deviation (σ) = (x - µ)2 / N

(σ) = σ2

Variance and Standard Deviation

Raw Score Mean Deviation Score Deviation Scores Squared x

x

(x - )

(x - )2

12 9 11 9 11 9 10 9 9 9 9 9 9 9 8 9 7 9 7 9 6 9

∑(x - )2 =

x x

x

Variance and Standard Deviation

Raw Score Mean Deviation Score Deviation Scores Squared x

x

(x - )

(x - )2

12 9 (12-9) = 3 11 9 (11-9) = 2 11 9 (11-9) = 2 10 9 (10-9) = 1 9 9 (9-9) = 0 9 9 (9-9) = 0 9 9 (9-9) = 0 8 9 (8-9) = -1 7 9 (7-9) = -2 7 9 (7-9) = -2 6 9 (6-9) = -3

∑(x - )2 =

x x

x

Variance and Standard Deviation

Raw Score Mean Deviation Score Deviation Scores Squared x

x

(x - )

(x - )2

12 9 3 (3)2 = 9 11 9 2 (2)2 = 4 11 9 2 (2)2 = 4 10 9 1 (1)2 = 1 9 9 0 (0)2 = 0 9 9 0 (0)2 = 0 9 9 0 (0)2 = 0 8 9 -1 (-1)2 = 1 7 9 -2 (-2)2 = 4 7 9 -2 (-2)2 = 4 6 9 -3 (-3)2 = 9

∑(x - )2 =

x x

x

Variance and Standard DeviationRaw Score Mean Deviation Score Deviation Scores Squared

x

x

(x - )

(x - )2

12 9 3 9 11 9 2 4 11 9 2 4 10 9 1 1 9 9 0 0 9 9 0 0 9 9 0 0 8 9 -1 1 7 9 -2 4 7 9 -2 4 6 9 -3 9

∑(x - )2 = 36

x x

x

Variance (s2 ) = (x – x )2 / n – 1 (s2 ) = 36 / 11 – 1 = 36 /10 = 3.6

Standard Deviation (s) = S2 (s) = 3.6 = 1.8973 = 1.90

n = 11

Coefficient of SkewnessTo measure the skew of the distribution, Pearson’s coefficient of skewness is often found

based on the relationship between the mean, median, and standard deviation

Can range in values from -3 to +3

Mean = 10Median = 7S = 4sk = 3(10 – 7)/4 = 3(3)/4 = 9/4 = 2.25 Data is positively skewed

For our data set Mean = 9Median = 9S = 1.90sk = 3(9 – 9)/1.90 = 3(0)/1.90 = 0/1.90 = 0Data is normally distributed

Coefficient of Skewness

Coefficient of SkewnessCoefficients that are zero or near zero will have data that will display equal tails

Coefficient of SkewnessCoefficients that are more positive in value will have data that displays a longer tail to the right

Coefficient of SkewnessCoefficients that are more negative in value will have data that displays a longer tail to the left