1. Displaying Quantitative Data Graphically and Describing It Numerically AP Statistics Chapters 4 & 5

2. Displaying Quantitative Data Histogram Stem-and-Leaf Plots Dotplots (Timeplots)

3. Histograms Bins and counts give the distribution of the quantitative data Bars touch—data is continuous Relative frequency histogram—useful and shows percentages, not counts

4. Stem-and-Leaf Plot Can see each individual data point Stem is like bin Might need to “split” 334779 Key: 3 4 = 34 46677789 5356777777 60001 799 2022222 2 577799999 Key: 2 4 = 24 3 44444 3667789999 42333344444 4 577779

5. Dotplot Useful in seeing how many individual data points in bin Good for small sets of data Not used too often

6. Describing a Distribution Whenever you are describing a distribution you need to describe it by the – Shape – Center – Spread – Any Unusual points (outliers, gaps)

7. Shape Is the shape? Uniform, Symmetric, Skewed How many modes (high points) – Unimodal, bimodal, multimodal

9. Center and Spread How we describe the center and spread of a distribution depends on the shape of the distribution.

10. Skewed Distribution Center: Median Spread: Interquartile Range (IQR) Both of these are “resistant” Both should include units

11. Skewed Distribution How to find the IQR 1. Find median 2. Find the median of both halves of data the lower median is 1 st Quartile the upper median is 3 rd Quartile 3. Subtract the two quartile scores ** 1 st Quartile = 25 th percentile ** 3 rd Quartile = 75 th percentile

12. Symmetric Distributions Center: Mean Spread: Standard Deviation Both are not “resistant” Both should include units

13. Standard Deviation Takes into account how far each value in a data set is from the mean Formula:

14. Properties of standard deviation 1.Only use with mean 2.If s = 0, there is no spread and all data pieces are same—otherwise s>0 and s gets larger as data pieces get more spread out. 3.A few outliers can really change the value of the standard deviation

15. Finding Standard Deviation by Hand Find the standard deviation: 10, 14, 15, 16, 20

16. Other information If distribution is symmetric, then mean=median If skewed right, mean>median If skewed left, mean<median Spread of distribution is just as important as the center How accurate: one or two decimal points more than original data

17. Distributions with Outliers Really just data that seems unusual Formally we compute fences and if data point is outside the fences, we consider it an outlier Always use common sense Upper fence: Lower fence:

18. Distributions with Outliers Tricky situation Since outliers affect mean and standard deviation, it is usually better to use median and IQR If the mean and median are not similar in value, report the median and IQR If the mean and the median are similar in value, report the mean and standard deviation. Sometimes (especially if the mean and median are not similar) it is a good idea to report your center and spread with and without the outlier and see what kind of effect removing the outlier has on the distribution.

19. Boxplots Complement histograms by providing more specific information Look at histogram and boxplot together Most useful when comparing distributions

20. Boxplots 5-Number Summary: Minimum, 1 st Quartile Score, Median, 3 rd Quartile Score, Maximum