Descriptive statistics

Describing data with numbers: measures of variability

What to describe?

• What is the “location” or “center” of the data?

• How do the data vary?

Measures of Variability

• Range

• Interquartile range

• Variance and standard deviation

• Coefficient of variation

All of these measures are appropriate for measurement data only.

Range

• The difference between largest and smallest data point.

• Highly affected by outliers.

• Best for symmetric data with no outliers.

What is the range?

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0

0

10

20

GPA

Fre

quency

GPAs of Spring 1998 Stat 250 Students

RangeDescriptive Statistics

Variable N Mean Median TrMean StDev SE MeanGPA 92 3.0698 3.1200 3.0766 0.4851 0.0506

Variable Minimum Maximum Q1 Q3GPA 2.0200 3.9800 2.6725 3.4675

Range = 3.98 - 2.02 = 1.96

Interquartile range

• The difference between the “third quartile” (75th percentile) and the “first quartile” (25th percentile). So, the “middle-half” of the values.

• IQR = Q3-Q1

• Robust to outliers or extreme observations.

• Works well for skewed data.

What is the Interquartile Range?

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0

0

10

20

GPA

Fre

quency

GPAs of Spring 1998 Stat 250 Students

Interquartile rangeDescriptive Statistics

Variable N Mean Median TrMean StDev SE MeanGPA 92 3.0698 3.1200 3.0766 0.4851 0.0506

Variable Minimum Maximum Q1 Q3GPA 2.0200 3.9800 2.6725 3.4675

IQR = 3.4675 - 2.6725 = 0.795

Variance

1n

2)x(x2s

1. Find difference between each data point and mean.

2. Square the differences, and add them up.

3. Divide by one less than the number of data points.

Variance

• If measuring variance of population, denoted by 2 (“sigma-squared”).

• If measuring variance of sample, denoted by s2

(“s-squared”).

• Measures average squared deviation of data points from their mean.

• Highly affected by outliers. Best for symmetric data.

• Problem is units are squared.

Standard deviation

• Sample standard deviation is square root of sample variance, and so is denoted by s.

• Units are the original units.

• Measures average deviation of data points from their mean.

• Also, highly affected by outliers.

What is the variance or standard deviation?

70 80 90 100 110 120 130 140 150 160Speed

Fastest Ever Driving Speed

126Women

100Men

226 Stat 100 Students, Fall '98

(MPH)

Variance or standard deviationSex N Mean Median TrMean StDev SE Mean female 126 91.23 90.00 90.83 11.32 1.01 male 100 06.79 110.00 105.62 17.39 1.74 Minimum Maximum Q1 Q3female 65.00 120.00 85.00 98.25male 75.00 162.00 95.00 118.75

Females: s = 11.32 mph and s2 = 11.322 = 128.1 mph2

Males: s = 17.39 mph and s2 = 17.392 = 302.5 mph2

What is the variance or standard deviation?

120 170 220 270

KPH

Fastest Ever Driving Speed

Sex

female

male

Variance or standard deviation

Sex N Mean Median TrMean StDev SE Mean female 126 152.05 150.00 151.39 18.86 1.68 male 100 177.98 183.33 176.04 28.98 2.90

Sex Minimum Maximum Q1 Q3female 108.33 200.00 141.67 163.75male 125.00 270.00 158.33 197.92

Females: s = 18.86 kph and s2 = 18.862 = 355.7 kph2

Males: s = 28.98 kph and s2 = 28.982 = 839.8 kph2

Coefficient of Variation

• Ratio of sample standard deviation to sample mean multiplied by 100.

• Measures relative variability, that is, variability relative to the magnitude of the data.

• Unitless, so good for comparing variation between two groups.

Coefficient of variation (MPH)Sex N Mean Median TrMean StDev SE Mean female 126 91.23 90.00 90.83 11.32 1.01 male 100 106.79 110.00 105.62 17.39 1.74 Minimum Maximum Q1 Q3female 65.00 120.00 85.00 98.25male 75.00 162.00 95.00 118.75

Females: CV = (11.32/91.23) x 100 = 12.4

Males: CV = (17.39/106.79) x 100 = 16.3

Coefficient of variation (KPH)Sex N Mean Median TrMean StDev SE Mean female 126 152.05 150.00 151.39 18.86 1.68 male 100 177.98 183.33 176.04 28.98 2.90

Sex Minimum Maximum Q1 Q3female 108.33 200.00 141.67 163.75male 125.00 270.00 158.33 197.92

Females: CV = (18.86/152.05) x 100 = 12.4

Males: CV = (28.98/177.98) x 100 = 16.3

The most appropriate measure of variability depends on …

the shape of the data’s distribution.

Choosing Appropriate Measure of Variability

• If data are symmetric, with no serious outliers, use range and standard deviation.

• If data are skewed, and/or have serious outliers, use IQR.

• If comparing variation across two data sets, use coefficient of variation.