AP Statistics 5 Number Summary and Boxplots Measures

AP Statistics 5 Number Summary and Boxplots

Measures of Center and Distributions n For a symmetrical distribution, the mean, median and the mode are the same.

Measure of Center and Distribution Continued n For skewed distributions, the mean is located closest to the skew.

Resistance A measure is said to be resistant if it is not very sensitive to the influence of extreme observations. n The mean is non-resistant. n The median is resistant. n

Range The range is a numerical summary. It is a crude measure of variation. n It is the difference between the largest and the smallest observation. n The range is extremely non-resistant. n

Interquartile Range (IQR) IQR is a number summary. It is a measure of variation. n IQR is a measure of the spread of the middle 50% (half) of data which is rank ordered. n

IQR Example 71, 76, 71, 87, 83, 70 are sample quiz scores. n Rank order: 70, 71, 71, 76, 83, 87 med = 71 n Left half: 70, 71 median = 71 (AKA Q 1) n Right half: 76, 83, 87 median = 83 (AKA Q 3) n This divides the distribution into quarters or “quartiles”. n

Five Number Summary The five number summary consists of the min, Q 1, median, Q 3, max. n It is a descriptor of a distribution. n It is found under 1 -var stats in the TI calculator. n In the previous example, the 5 number summary was 70, 71, 83, 87 n The IQR of the data is calculated: n IQR = (Q 3 - Q 1) = (83 -71) = 12 n

Boxplots - Example

Identify Outliers – Modified Boxplots Modify boxplots to expose extremes. n The new low end is Q 1 – 1. 5(IQR) n The new high end is Q 3 + 1. 5(IQR) n

Example n The five number summary is: 24, 70, 85. 5, 89, 97

Example Continued IQR = (Q 3 - Q 1) = (89 – 70) = 19. n Lower limit = Q 1 – (1. 5 · IQR) = 70 - 1. 5 · 19 = 41. 5 n Upper limit = Q 3 + (1. 5 · IQR) = 89 + 1. 5 · 19 = 117. 5 n Since 24 lies outside the lower and upper limit, it is a potential outlier. There may be more, but we don’t have the original data. n

Example Continued

Homework n Worksheet.