Stat 20: Intro to Probability and Statistics

Transcription

1 Stat 20: Intro to Probability and Statistics Lecture 5: Summary Statistics Tessa L. Childers-Day UC Berkeley 30 June 2014

2 By the end of this lecture... You will be able to: Describe a data set by its: Shape Location Spread Describe a sketched histogram or smoothed histogram using statistical language 2 / 27

3 Recap Last time: ways to display quantitative data Stem and Leaf Plot Scatterplot Histogram Frequency Density Smoothed Histogram All of these display a data set. But how do we describe the displays (and thus the data set)? 3 / 27

4 How are these shapes different? Similar? 4 / 27

5 How are these shapes different? Similar? (cont.) 5 / 27

6 How are these shapes different? Similar? (cont.) Both are symmetric Light/narrow tails Heavy/wide tails 6 / 27

7 How are these shapes different? Similar? (cont.) Both are skewed (not symmetric) Right tailed/positively skewed Left tailed/negatively skewed 7 / 27

8 How are these shapes different? Similar? (cont.) We use symmetry, tail heaviness, and skewness to describe the shape of the histogram. But what about other differences? Shapes are the same. What s different? 8 / 27

9 Where is the center? Measure of Location = Measure of the Center 9 / 27

10 Where is the center? (cont.) Pretty clear that center is at 0 What about here? 10 / 27

11 What is the center? Need to know how to define the center. 3 popular ways 1 Mean (average) 2 Mode 3 Median Summary statistic(s): numbers meant to summarize the data 11 / 27

12 The Mean/Average Say we have a list of numbers, call it x: i x i n = length of the list = 4. The average is the sum of all the entries in a list average = the number of entries in the list n i=1 = x 4 i i=1 = x i n 4 = = = 6 12 / 27

13 The Mode Say we have a list of numbers, call it x: n = length of the list = 4. i x i The mode is the number that occurs most frequently. If two (or more) numbers occur with the same frequency, then we have a bimodal (or multimodal) list. Here, mode = / 27

14 The Median Say we have a list of numbers, call it x: i x i n = length of the list = 4. Sort the list. If n is odd: median = middle number in sorted list If n is even: median = avg of two middle numbers in sorted list Here, median = / 27

15 Comparing Mean, Mode, Median If the histogram or list is symmetric, they are the same Mean, Median, Mode / 27

16 Comparing Mean, Mode, Median (cont.) If the histogram or list is not symmetric, they are not the same Mode Median Mean Median Mean Mode / 27

17 Percentiles Say we have a list of numbers, call it x: i x i n = length of the list = 4. Sort the list (or make a histogram). j th percentile = value so that j% of list (or histogram) is less than or equal to that value 10 th percentile = value so that 10% of list (or histogram) is less than or equal to that value 95 th percentile = value so that 95% of list (or histogram) is less than or equal to that value 50 th percentile = value so that 50% of list (or histogram) is less than or equal to that value 17 / 27

18 Percentiles (cont.) i x i % of list = 25% n =.25 n =.25 4 = 1 25 th percentile = value so that 1 number is less than or equal to it = 4 50% of list = 50% n =.50 n =.50 4 = 2 50 th percentile = value so that 2 numbers are less than or equal to it = 6 75% of list = 75% n =.75 n =.75 4 = 3 75 th percentile = value so that 3 numbers are less than or equal to it = 6 18 / 27

19 Percentiles (cont.) Histogram of x % 50% 25% 25% of histogram is below th percentile = % of histogram is below 6 50 th percentile = 6 75% of histogram is below th percentile = x The histogram summarizes data, the list does not 19 / 27

20 How are these shapes different? Similar? We use mean, mode, median, percentiles to describe the location of the histogram. But what about other differences? Density (% per unit x) Shapes and locations are the same. What s different? 20 / 27

21 What is the spread? Density (% per unit x) Measure of Spread = Measure of Distance Around Center 21 / 27

22 RMS RMS = Root-Mean-Square = Square each item, find the mean, take the square root i x i xi Mean(x 2 i ) = n i=1 x 2 i n 4 i=1 = x i 2 4 = = = 38 RMS = n i=1 x 2 i n = 38 = / 27

23 Standard Deviation SD = Standard Deviation = RMS of deviations from mean. Recall our mean is 6 i x i x i mean (x i mean) n Mean[(x i mean) 2 i=1 ] = (x i mean) 2 n 4 i=1 = (x i mean) 2 4 = = 8 4 = 2 n i=1 SD = (x i mean) 2 n = 2 = / 27

24 Standard Deviation (cont.) Another calculation: i x i xi SD = Mean of Squares Squared Mean = MS SM ( n i=1 = x i 2 ) ( n i=1 x ) 2 i n n (16 ) ( ) = = 4 62 = = 2 = / 27

25 Standard Deviation (cont.) The SD is in the same units as the data The SD is always positive For most, but not all lists, About 68% of the data are within 1 SD of the mean (in the range [mean - SD, mean + SD]) About 95% of the data are within 2 SDs of the mean (in the range [mean - 2 SD, mean + 2 SD] Density (% per unit x) Mean Mean SD Mean + SD Mean 2xSD Mean + 2xSD / 27

26 IQR IQR = Interquartile Range = 75 th percentile - 25 th percentile The IQR is in the same units as the data The IQR is always positive For all lists, The IQR is the number of units which contains 50% of the data The range [25 th percentile, 75 th percentile] contains the middle 50% of the data Density (% per unit x) Median 25th Percentile 75th Percentile / 27

27 Important Takeaways Need to summarize data use data displays or summary statistics Describing Data Displays Symmetry Tails Skew Summary Statistics Location: mean, mode, median, percentiles Spread: RMS, SD, IQR Next time: Normal Curve/Approximation 27 / 27