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Chapter 5

Chapter 5. Describing Distributions Numerically. Definitions. Mean : the simple average you learned how to do in 6 th grade Median : using a list arranged in order from low to high, the middle number Mode : the value that occurs most often

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Chapter 5

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  1. Chapter 5 Describing Distributions Numerically

  2. Definitions • Mean: the simple average you learned how to do in 6th grade • Median: using a list arranged in order from low to high, the middle number • Mode: the value that occurs most often • Midrange: the average of the minimum and maximum values • Very sensitive to skewed distributions and outliers.

  3. Mode • The most frequent value in the data set • Ex: Here is a list with the number of years each employee has worked at a company, what is the mode? 1 2 2 4 4 4 5 5 5 5 5 5 5 7 10 11

  4. Median: The Middle of Everything • The median is the value with exactly half the data values below it and half above it. • It is the middle data • The average of the two numbers in the middle if its an even number of values

  5. Mean • To find the mean add up all the values and divide by the total number of cases • Ex: the total number of boxes sold between January and June are: 249, 337, 163, 289, 298, and 104. How many boxes were sold on average?

  6. Mean or Median? • In symmetric distributions, the mean and median are approximately the same in value, so either measure of center may be used. • For skewed data, though, it’s better to report the median than the mean as a measure of center.

  7. Range • Always report a measure of spread along with a measure of center when describing a distribution numerically. • The range of the data is the difference between the maximum and minimum values: Range = max – min • A disadvantage of the range is that a single extreme value can make it very large and, thus, not representative of the data overall.

  8. Spread: The Interquartile Range • The interquartile range (IQR) lets us ignore extreme data values and concentrate on the middle of the data. • To find the IQR, we first need to know what quartiles are…

  9. Spread: The Interquartile Range (cont.) • Quartiles divide the data into four equal sections. • The lower quartile is the point that 1 quarter of the data lies below • The upper quartile is the point that 1 quarter al the data lies above • The difference between the quartiles is the IQR, so IQR = upper quartile – lower quartile

  10. Spread: The Interquartile Range (cont.) • The lower and upper quartiles are the 25th and 75thpercentiles of the data, so… • The IQR (purple) contains 50% of the values of the distribution

  11. The Five-Number Summary • The five-number summary of a distribution reports its median, quartiles, and extremes (maximum and minimum). • Example: The five-number summary for the ages at death for rock concert goers who died from being crushed is

  12. Finding the IQR by hand • Sort the values from smallest to largest • Split the data into 2 halves at the median • When N is odd, include the median in both halves • Q1: the median of the lower half of the data • Find the median of just the lower half of the data • Q3: the median of the upper half of the data • Find the median of just the upper half of the data • Remember: • Lower Quartile = 25th percentile = Q1 • Median = 50th percentile = Q2 • Upper Quartile = 75th percentile = Q3

  13. Example Chapter 5 #4 • Here are the annual numbers of deaths from tornados in the United States from 1990 through 2000: 53, 39, 39, 33, 69, 30, 25, 67, 130, 94, 40 Find the Mean, Median, Quartiles, Range and IQR • Mean: (53 + 39+…+40)/11 = 56.3 • Median: 40 • Q1 Median of lower half: (33 + 39)/2 = 36 • Q3 Median of upper half: (67 + 69)/2 = 68 • Range: 130 – 25 = 105 • IQR Q3 – Q1: 68 – 36 = 32

  14. Boxplots • A boxplot is a graphical display of the five-number summary. • Boxplots are particularly useful when comparing groups.

  15. Constructing Boxplots • The Crowdsafe Database lists the ages, names, causes, and locations of these unfortunate concert-goers. During the period of 1999 and 2000 there were 66 people who died from “crowd crush.” Here is a 5 number summary of their ages:

  16. Constructing Boxplots Draw a single vertical axis spanning the range of the data (10 to 50). Draw short horizontal lines at the lower (17) and upper quartiles (22) and at the median (19). Then connect them with vertical lines to form a box.

  17. Constructing Boxplots (cont.) Erect “fences” around the main part of the data. The upper fence is 1.5 IQRs (22-17 =5) above the upper quartile 22+ 1.5(5) = 29.5 The lower fence is 1.5 IQRs below the lower quartile 17 – 1.5(5) = 9.5 Note: the fences only help with constructing the boxplot and should not appear in the final display.

  18. Constructing Boxplots (cont.) Use the fences to grow “whiskers.” Draw lines from the ends of the box up and down to the most extreme data values found within the fences. We Don’t have all the data points so lets just say its 28 and 13 If a data value falls outside one of the fences, we do not connect it with a whisker.

  19. Constructing Boxplots (cont.) Add the outliers by displaying any data values beyond the fences with special symbols. Our outliers are 37 and 47 We often use a different symbol for “far outliers” that are farther than 3 IQRs from the quartiles (3 x 5).

  20. Constructing Boxplots (cont.) • Compare the histogram and boxplot for rock concert deaths:

  21. Comparing Groups With Boxplots • The following set of boxplots compares the effectiveness of various coffee containers: • What does this graphical display tell you?

  22. Summarizing Symmetric Distributions • The distribution of pulse rates for 52 adults is generally symmetric, with a mean of 72.7 beats per minute (bpm) and a median of 73 bpm:

  23. What About Spread? The Standard Deviation • A more powerful measure of spread than the IQR is the standard deviation, which takes into account how far each data value is from the mean. • A deviation is the distance that a data value is from the mean. • Since adding all deviations together would total zero, we square each deviation and find an average of sorts for the deviations.

  24. The variance, notated by s2, is found by summing the squared deviations and (almost) averaging them: The variance will play a role later in our study, but it is problematic as a measure of spread—it is measured in squared units! What About Spread? The Standard Deviation (cont.)

  25. The standard deviation, s, is just the square root of the variance and is measured in the same units as the original data. What About Spread? The Standard Deviation (cont.)

  26. Standard Deviation Ex. (81) • Suppose we are given the values: 4, 3, 10, 12, 8, 9, and 3 • The mean, γ = 7

  27. Standard Deviation Example (cont.) • Add up the squared deviations: ∑: 9 + 16 + 9 + 25 + 1 + 4 + 16 = 80 • Divide by n – 1: 80/6 = 13.33 • Finally take the square root: √13.33 = 3.65

  28. Using your calculator TI 83/84 • STAT • 1 or Enter • Lists functions L1 to L6 • Enter your data values • STAT • CALC • Enter • 1 or Enter (1-Var STATS) • Press 2nd button and L1 • Enter TI 89 • APPS • 6: Data Matrix Editor • 1 and Enter • C1: list your data values • F5 Calc: OneVar Enter • X: type C1 • Enter • Enter

  29. Using your calculator (cont.) TI 83/84 Printout TI 89 Printout

  30. Thinking About Variation • Since Statistics is about variation, spread is an important fundamental concept of Statistics. • Measures of spread help us talk about what we don’t know. • When the data values are tightly clustered around the center of the distribution, the IQR and standard deviation will be small. • When the data values are scattered far from the center, the IQR and standard deviation will be large.

  31. Shape, Center, and Spread • When telling about a quantitative variable, always report the shape of its distribution, along with a center and a spread. • If the shape is skewed, report the median and IQR. • If the shape is symmetric, report the meanand standard deviationand possibly the median and IQR as well.

  32. What About Outliers? • If there are any clear outliers and you are reporting the mean and standard deviation, report them with the outliers present and with the outliers removed. The differences may be quite revealing. • Note: The median and IQR are not likely to be affected by the outliers.

  33. What Can Go Wrong? • Don’t forget to do a reality check—don’t let technology do your thinking for you. • Sort the values before finding the median or percentiles. • Don’t compute numerical summaries of a categorical variable. • Beware of outliers • Make a picture (make a picture and make a picture)

  34. What Can Go Wrong? (cont.) • Be careful when comparing groups that have very different spreads. • Consider these side-by-side boxplots of cotinine levels:

  35. *Re-expressing to Equalize the Spread of Groups • Here are the side-by-side boxplots of the log (cotinine) values:

  36. For Next week • Homework Chapter 5: 3, 7, 9, 12, 27 a-c, 28 • Extra Credit is due (5 points) • Quiz 1: Closed Notes: • Know the difference between categorical and quantitative variables • Calculate the mean, median, and mode • Find percents from a contingency table • Sketch a stem and leaf plot

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