# Box Plots

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## Box Plots

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1. Box Plots

2. Mean ( x ): Statistical Measures Statistics practice of analyzing a set of data Measures of Central Tendency: numbers that represent the middle of the data (mean, median, mode) Arithmetic average Median: Middle of the data listed in ascending order (use if there is an outlier) Mode: Most common number (if modes – can be more than one) How much data is spread out Variance: Standard Deviation(σ): Measure of variation (high=spread out) Skewed Right (positively): Less data to the right. Less data to the left Skewed Left (negatively):

3. Middle 50% 5 Number Summary: Min: Minimum Value (0 Percentile) Q1: Quartile 1 (25th Percentile) Med (Q2): Median (50th Percentile) Q3: Quartile 3 (75th Percentile) Max: Maximum or Q4 (100th Percentile) 25% 25% 25% 25% Min Med Max Q1 Q2 Q3 Q4 IQR: Q3 – Q1 (where the middle 50% are)

4. Calculator Commands • Enter data into STATEDIT • STAT→CALC → 1-Var STATS… Mean Standard Deviation Minimum 1st Quartile Median 3rd Quartile Maximum Calculator Information You still have to find the mode by looking for the most common number(s), calculate the IQR by finding the diffference of Q3 and Q1 and finding the range by finding the difference of maximum and minimum

5. Listed below are the weights of 10 people (in pounds) 130, 150, 160, 145, 142, 143, 170, 132, 145, 156 Make the 130 a 120 and the 156 a 166. Recalculate What changed? Why? Find the mean: mode: standard deviation median: minimum: 1st quartile: 3rd quartile: maximum IQR: Range: Make a box plot: Skewed: 147.3 145 11.62 145 130 142 156 170 Q3-Q1: 156-142=14 Max-min: 170-130=40 Right (positive) Standard deviation, Range The data is more spread out

6. Class Data set of “The day of the month you were born on” Find the mean: mode: standard deviation median: minimum: 1st quartile: 3rd quartile: maximum IQR: Range: Make a box plot: Skewed:

7. The following is the amount of black M&M’s in a bag: 12, 13,14, 15, 15, 16, 17, 20, 21,22,23,24,25 Find the mean and standard deviation Mean: 18.23 Standard Deviation: 4.28 The following is the amount of black M&M’s in a bag: 9, 10,11, 14, 15, 16, 17, 20, 21,23,26,27,28 Find the mean and standard deviation [Default] [MC Any] [MC All] Mean: 18.23 Standard Deviation: 4.28 Explain why the means are the same but the standard deviation is larger for the 2nd example. The data is more spread out although it’s the same average.

8. Test Scores (n=60) 60*0.25 = 15 25% 25% 25% 25% 15 15 15 15 70-89 Between what scores do the middle 50% lie? Between what scores does the lowest 25% lie? Which range of scores has more density? (more numbers in a smaller number) 4. Estimate how many people got between 85-89? 5. Estimate how many people got below an 85? 6. What is the IQR? 7. What percentile did a person with a 70 get? 55-70 85-89 15 30 89-70 = 19 25

9. Box plot of 80 Bowlers 25% 25% 80*.25=20 25% 25% 20 20 20 20 60 70 80 90 100 110 120 130 140 145 What is the maximum score? What is the IQR? What percentage of bowlers got above a 85? How many bowlers got below a 100? What percentile did a 120 get? Between what scores did the top 25% get? Where are there less density of bowlers? 140 120-85=35 25+25+25=75 20 + 20 = 40 75% (75% are below) 120-140 60-85

10. Which date has the most variability (spread)? Make a statement about airline delays 3) Which would you expect to have larger variability (IQR) A) Heights of freshmen B) Ages of freshmen C) Weights of freshmen D) Heights of 5 year-olds?

11. [Which of the following will have the most variability? • [Heights of people in this room] • [Ages of people in this room] • [The number of countries that people have been to in this room?] [Default] [MC Any] [MC All] Variability: How close the numbers are together More spread out= HighVariability = Large Standard Deviation =High IQR

12. Which would have a lower standard deviation? (Be prepared to explain): • [The heights of students in this class] • [The heights of students in this school] [Default] [MC Any] [MC All]

13. Normal Distribution Bell Curve http://en.wikipedia.org/wiki/File:Standard_deviation_diagram.svg http://en.wikipedia.org/wiki/Skewness

14. Determine if the following examples are Normally Distributed, Positively Skewed, or Negatively Skewed.

15. Place the following under negatively skewed, normally distributed, or positively skewed, or random? • The amount of chips in a bag • The sum of the digits of random 4-digit numbers? • The number of D1’s that students in this class have gotten? • The weekly allowance of students • Age of people on a cruise this week F) The shoe sizes of females in this class

16. Debate: • Side 1) You are trying to convince your teacher to always curve test grades to a standard deviation • Side 2) You are trying to convince your teacher to never curve test grades to a standard deviation

17. The next few slides are done as a wrap up or warm-up the next few daysThere are three at the end for easy questions

18. Mode: Most often number. Mean: Average. Median: The middle number when arranged from smallest to largest. Best to show when there are outliers!!! 1) Find the mode, mean, and median: 5,7,9,9,30 9 12 9 2) Which is the largest? Mean 3) Now include a 90 in the data. Which of the three changed the most? Mean: It went from 12 to 25 4) When they list salaries, why do they state the median price and not the mean price? Median is less affected by outliers

19. Stats for Winston-Salem stats (Rated #10 as best place to live by Money Magazine)

20. Deeper Understanding • Suppose there are 20 tests and the scores are all an 80%. What would change if 2 more tests were added that were both a 90%, mean or median? • What if there were 20 tests, 4 were 70%, 12 were 80%, and 4 were 90%. Three more tests were added to group scoring 70%, 90%, and 100%. How would the mean or median change?

21. Trick or Treat • Ten neighborhood kids went out to get candy. Here is a list of the number of treats they received: 45, 34, 56, 32, 10, 32, 62, 11, 55, 34 • Find the mean, median, and IQR of the treats. • The kid who got 62 treats, went back out and got 262 treats. Find the new mean, median and IQR. • Which does a better job of describing the typical number of treats for the new data? Why? • Draw a box plot.

22. Which way is the data skewed? Suppose you wanted to open restaurants. Where would you put a few of them? Where would you want a lot of them?

23. Ex. Find the mean, median, and mode of 85, 76, 88, 91, 85, 58, 88, 91, 97, 91, 88, 97, 97 • Mean ( ) = median = , mode = Draw a box plot of the data.

24. FIVE-NUMBER SUMMARY: Find the 5 number summary and draw a box plot. Maria: 8, 9, 6, 7, 9, 8, 8, 6, 9, 9, 8, 7, 8, 7, 9, 9, 7, 7, 8, 9 6 Min.: Q1: 7 8 Q2 (median): 9 Q3: Max.: 9 Interquartile Range (IQR): Q3 – Q1= 9 – 7 = 2 6 7 8 9

25. FIVE-NUMBER SUMMARY: Find the 5 number summary and draw a box plot. Gia: 8, 9, 9, 9, 6, 9, 8, 6, 8, 6, 8, 8, 8, 6, 6, 6, 3, 8, 8, 9 Min.: 3 Q1: 6 3 9 8 6 Q2 (median): 8 8.5 Q3: 8.5 Max.: 9 8.5 – 6 = 2.5 Interquartile Range (IQR): Q3 – Q1=