Organizing Data AP Stats Chapter 1

Organizing Data Categorical Categorical Dotplot (also used for quantitative) Dotplot (also used for quantitative) Bar graph Bar graph Pie chart Pie chart Quantitative Quantitative Stemplots Stemplots Unreasonable with large data sets Unreasonable with large data sets Histogram Histogram Frequency/relative frequency Frequency/relative frequency

Describing Distributions Remember “SECS” Remember “SECS” S – Shape S – Shape E – Extreme Values (outliers) E – Extreme Values (outliers) C – Center C – Center S – Spread S – Spread

Shape Symmetric Symmetric Values smaller and larger than the midpoint are mirror images. Values smaller and larger than the midpoint are mirror images. Skewed Skewed The tail on one end is much longer than the other tail. The tail on one end is much longer than the other tail.

Example: Symmetric

Examples: Skewed

Ways to Measure Center Mean Mean The mean is not a resistant measure of center. (sensitive to outliers) The mean is not a resistant measure of center. (sensitive to outliers) Used mostly with symmetric distributions. Used mostly with symmetric distributions.

Ways to measure center Median Median Midpoint of a distribution Midpoint of a distribution Median is a resistant measure of center Median is a resistant measure of center Used with symmetric or skewed distributions. Used with symmetric or skewed distributions.

Ways to Measure Spread 1) Range 1) Range Highest value – lowest value Highest value – lowest value Problem: could be based on outliers Problem: could be based on outliers 2) Quartiles (for use with median) 2) Quartiles (for use with median) pth percentile – value such that p percent of the observations fall at or below it pth percentile – value such that p percent of the observations fall at or below it Q 1 (quartile 1): 25 th percentile Q 1 (quartile 1): 25 th percentile Median of the first half of the data Median of the first half of the data Q 3 (quartile 3): 75 th percentile Q 3 (quartile 3): 75 th percentile Median of the second half of the data Median of the second half of the data

Ways to Measure Spread 5 Number Summary 5 Number Summary Minimum, Q 1, median, Q 3, maximum Minimum, Q 1, median, Q 3, maximum The 5-number summary for a distribution can be illustrated in a boxplot. The 5-number summary for a distribution can be illustrated in a boxplot.

1.5 x IQR Rule for Outliers IQR = Q 3 – Q 1 (Interquartile Range) IQR = Q 3 – Q 1 (Interquartile Range) Rule: If an observation falls more than 1.5 x IQR above Q 3 or below Q 1, then we consider it an outlier. Rule: If an observation falls more than 1.5 x IQR above Q 3 or below Q 1, then we consider it an outlier. The 5 Number Summary can be used for distributions which are skewed, or which have strong outliers. The 5 Number Summary can be used for distributions which are skewed, or which have strong outliers.

Ways to Measure Spread Standard deviation (for use with the mean) Standard deviation (for use with the mean) Std Dev tells you, on average, how far each observation is from the mean. Std Dev tells you, on average, how far each observation is from the mean.

Properties of Standard Deviation s gets larger as the data become more spread out. s gets larger as the data become more spread out. Only use mean and std dev for reasonably symmetric distributions which are free of outliers. Only use mean and std dev for reasonably symmetric distributions which are free of outliers.

Linear Transformation of Data X new = a + bx X new = a + bx The shape of the distribution does not change. The shape of the distribution does not change. Multiplying each observation by a positive number, b, multiplies both measures of center and measures of spread by b. Multiplying each observation by a positive number, b, multiplies both measures of center and measures of spread by b. Adding the same number, a, to each observation adds a to measures of center and to quartiles, but does not change measures of spread. Adding the same number, a, to each observation adds a to measures of center and to quartiles, but does not change measures of spread.