Organizing Data AP Stats Chapter 1 Organizing Data Categorical Dotplot (also used for quantitative) Bar graph Pie chart Quantitative Stemplots Unreasonable with large data sets Histogram Frequency/relative frequency Describing Distributions Remember “SECS-C” S – Shape E – Extreme Values (outliers) C – Center S – Spread C – Context **Make meaningful descriptions and comparisons. Don’t just list numbers.** Shape Symmetric Values smaller and larger than the midpoint are mirror images. Skewed The tail on one end is much longer than the other tail. Example: Symmetric Examples: Skewed Ways to Measure Center Mean x = x1 + x2 + . . . . + xn n xi x = n The mean is not a resistant measure of center. (sensitive to outliers) Used mostly with symmetric distributions. Ways to measure center Median Midpoint of a distribution Median is a resistant measure of center Used with symmetric or skewed distributions. Ways to Measure Spread 1) Range Highest value – lowest value Problem: could be based on outliers 2) Quartiles (for use with median) pth percentile – value such that p percent of the observations fall at or below it Q1 (quartile 1): 25th percentile Median of the first half of the data Q3 (quartile 3): 75th percentile Median of the second half of the data Ways to Measure Spread 5 Number Summary Minimum, Q1, median, Q3, maximum The 5-number summary for a distribution can be illustrated in a boxplot. 1.5 x IQR Rule for Outliers IQR = Q3 – Q1 (Interquartile Range) Rule: If an observation falls more than 1.5 x IQR above Q3 or below Q1, then we consider it an outlier. 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) Std Dev tells you, on average, how far each observation is from the mean. variance: s2 = std dev: s = (xi - x)2 n-1 (xi - x)2 n-1 Properties of Standard Deviation 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. Linear Transformation of Data Xnew = a + bx 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. Adding the same number, a, to each observation adds a to measures of center and to quartiles, but does not change measures of spread.