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Descriptive statisticsDescriptive^ statisticsDescribing data with numbers:^ measures of variability

y

What to describe?• What is the “location” or “center” of thedata?• How do the data vary?

Range• The difference between largest and smallestdata point.p • Highly affected by outliers.• Best for symmetric data with no outliers• Best for symmetric data with no outliers.

What is the range? GPA^ f S^ i^ 1998 St t 250 St d

t GPAs of Spring 1998 Stat 250 Students (^20) ency (^1010) Freque 2.0^ 2.2^ 2.4^ 2.6^ 2.8^ 3.0^ 3.2^ 3.^ 0 3.6^ 3.8^ 4.0 GPA

Interquartile range• The difference between the “ thirdquartile ” (75th percentile) and the “

first

q^ (^ p^

quartile ” (25th percentile). So, the“middle-half” of the values.• IQR = Q3-Q1• Robust to outliers or extreme observations• Robust to outliers or extreme observations.• Works well for skewed data.

What is the Interquartile Range?^ GPA^ f S^ i^ 1998 St t 250 St d

t GPAs of Spring 1998 Stat 250 Students (^20) ency (^1010) Freque 2.0^ 2.2^ 2.4^ 2.6^ 2.8^ 3.0^ 3.2^ 3.^ 0 3.6^ 3.8^ 4.0 GPA

Variance^ 1. Find difference between^ each data point and mean. 2 )x(x−2. Square the differences, and ∑ 2 s =add them up.p 1 n− 1 n 3. Divide by one less than thenumber of data points.

Variance• If measuring variance of population, 2 denoted by σ(“ sigma-squared ”).y (^ g^ q^ ) • If measuring variance of sample, denoted by 2 s (“ s-squared ”) s ( s squared ). • Measures average squared deviation of datapoints from their meanpoints from their mean.• Highly affected by outliers. Best fori dsymmetric data.• Problem is units are squared.

What is the varianceWhat^ is the varianceor standard deviation?^ Fastest Ever Driving Speed226 Stat 100 Students Fall '98226 Stat 100 Students, Fall 98 100 Men 126 Women^70 80 90 100 110 120

(^140 150 160) Speed Women

(MPH)

Variance or standard deviation Sex^ N^ Mean^ Median

TrMean^ StDev^ SE^ Mean female^126 91.23^ 90.

90.83^ 11.32^ 1.

male^100 06.79^ 110.

105.62^ 17.39^ 1.74Minimum Maximum^ Q1^ Q female^ 65.00^ 120.

85.00^ 98.

male^ 75.00^ 162.

95.00^ 118.

Females: s = 11.32 mph and s

2 2 2 = 11.32= 128.1 mph

2 Males: s = 17.39 mph and s

2 2 = 17.39= 302.5 mph

Variance or standard deviation Sex N^ Mean^ Median

TrMean^ StDev^ SE^ Mean female^126 152.05^ 150.

151.39^ 18.86^ 1.

male^100 177.98^ 183.

176.04^ 28.98^ 2.

Sex^ Minimum^ Maximum

Q1^ Q

female^ 108.33^ 200.

141.67^ 163.

male^125 00 270

male^ 125.00^ 270.

158.33^ 197.92^2 2 2 Females: s = 18.86 kph and s^ = 18.86= 355.7 kph^2 2 2 Males: s = 28.98 kph and s^ = 28.98= 839.8 kph

Coefficient of Variation• Ratio of sample standard deviation tosample mean multiplied by 100.p^ p^ y • Measures^ relative variability

, that is,

variability relative to the magnitude of thevariability relative to the magnitude of thedata.• Unitless so good for comparing variation• Unitless, so good for comparing variationbetween two groups.

Coefficient of variation (KPH) S N^ M^ M^ di^

T^ M^ StD^ SE^ M Sex^ N^ Mean^ Median

TrMean^ StDev^ SE^ Mean female^126 152.05^ 150.

151.39^ 18.86^ 1.

male^100 177.98^ 183.

176.04^ 28.98^ 2.

Sex^ Minimum^ Maximum

Q1^ Q

female^ 108.33^ 200.

141.67^ 163.

male^ 125.00^ 270.

158.33^ 197.

male^ 125.00^ 270.

158.33^ 197.92 Females: CV = (18.86/152.05) x 100 = 12.4Males: CV = (28.98/177.98) x 100 = 16.

The most appropriate measureof variability depends onof variability depends on …the shape of the data’sdistribution.