AP Statistics 5 Number Summary and Boxplots

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

Measure of Center and Distribution Continued For skewed distributions, the mean is located closest to the skew. 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. A measure is said to be resistant if it is not very sensitive to the influence of extreme observations. The mean is non-resistant. The mean is non-resistant. The median is resistant. The median is resistant.

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

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

IQR Example 71,71,76,71,87,83,70 are sample quiz scores. 71,71,76,71,87,83,70 are sample quiz scores. Rank order: 70, 71, 71, 71, 76, 83, 87 med = 71 Rank order: 70, 71, 71, 71, 76, 83, 87 med = 71 Left half: 70, 71, 71 median = 71 (AKA Q1) Left half: 70, 71, 71 median = 71 (AKA Q1) Right half: 76, 83, 87 median = 83 (AKA Q3) Right half: 76, 83, 87 median = 83 (AKA Q3) This divides the distribution into quarters or “quartiles”. This divides the distribution into quarters or “quartiles”. Min, Q1, Median, Q3, Max Min, Q1, Median, Q3, Max

Five Number Summary The five number summary consists of the min, Q1, median, Q3, max. The five number summary consists of the min, Q1, median, Q3, max. It is a descriptor of a distribution. It is a descriptor of a distribution. It is found under 1-var stats in the TI calculator. It is found under 1-var stats in the TI calculator. In the previous example, the 5 number summary was 70, 71, 71, 83, 87 In the previous example, the 5 number summary was 70, 71, 71, 83, 87 The IQR of the data is calculated: The IQR of the data is calculated: IQR = (Q3 - Q1) = (83 -71) = 12 IQR = (Q3 - Q1) = (83 -71) = 12

Boxplots - Example

Identify Outliers – Modified Boxplots Modify boxplots to expose extremes. Modify boxplots to expose extremes. The new low end is Q1 – 1.5(IQR) The new low end is Q1 – 1.5(IQR) The new high end is Q (IQR) The new high end is Q (IQR)

Example The five number summary is: 24, 70, 85.5, 89, 97 The five number summary is: 24, 70, 85.5, 89, 97

Example Continued IQR = (Q3 - Q1) = (89 – 70) = 19. IQR = (Q3 - Q1) = (89 – 70) = 19. Lower limit = Q1 – (1.5 · IQR) = · 19 = 41.5 Lower limit = Q1 – (1.5 · IQR) = · 19 = 41.5 Upper limit = Q3 + (1.5 · IQR) = · 19 = Upper limit = Q3 + (1.5 · IQR) = · 19 = 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. 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.

Example Continued

Homework Worksheet. Worksheet.