1. Interpreting data … Drawing and comparing Box and Whisker diagrams (Box plots)

2. Learning objectives All: (B grade) Calculate quartiles and draw box and whisker diagrams Most: (A grade) Interpret box and whisker diagrams and use to compare datasets Some: (Stats GCSE) Use all terminology and use IQR to find “outliers”

3. A list of data The weights (KG) of 15 children: 37, 42, 31, 35, 48, 29, 50, 36, 44, 28, 63, 35, 41, 52, 43 Difficult to UNDERSTAND what these children look like from the list …

4. Minimum = 28KG Maximum = 63KG Range = 35KG Mode = 35KG Median = 41KG Mean = 40.9KG The weights (KG) of 15 children: 28, 29, 31, 35, 35, 36, 37, 41, 42, 43, 44, 48, 50, 52, 63

5. Stem and leaf … 28, 29, 31, 35, 35, 36, 37, 41, 42, 43, 44, 48, 50, 52, 63 2 8 9 3 1 5 5 6 7 4 1 2 3 4 8 5 0 2 6 3 Key: 2 9 means 29 ORDERED STEM & LEAF

6. Another useful summary A diagram to show: min (28KG), max (63KG), median (41KG) … MinMedian Max

7. Median ½(n + 1) th piece of data (ordered) 28, 29, 31, 35, 35, 36, 37, 41, 42, 43, 44, 48, 50, 52, 63 15 items of data … n = 15 ½(n + 1) = ½(15 + 1) = 8 th item

8. Lower Quartile ¼(n + 1) th piece of data (ordered) 28, 29, 31, 35, 35, 36, 37, 41, 42, 43, 44, 48, 50, 52, 63 15 items of data … n = 15 ¼(n + 1) = ¼(15 + 1) = 4 th item

9. Upper Quartile ¾(n + 1) th piece of data (ordered) 28, 29, 31, 35, 35, 36, 37, 41, 42, 43, 44, 48, 50, 52, 63 15 items of data … n = 15 ¾(n + 1) = ¾(15 + 1) = 12 th item

10. Add that to our box plot A diagram to show: min (28KG), lower quartile = 35KG max (63KG), upper quartile = 48KG median (41KG) … Min Median Max LQ UQ

11. Some terminology Min Median Max LQ UQ Q0Q0 Q1Q1 Q2Q2 Q3Q3 Q4Q4 Alternative names for quartiles

12. Some terminology UQ – LQ = Interquartile Range (IQR) Max – Min = Range

13. Some terminology Positive skew: median closer to LQ than UQ Negative skew: median closer to UQ than LQ Symmetrical distribution

14. Interpreting the box plot Easily see lightest / heaviest and range The ‘box’ contains the middle 50% of people (the most ‘representative half’) The ‘whiskers’ show the lightest 25% and heaviest 25% of people (extremes)

15. Comparing groups Boys Girls “Lightest girl lighter than lightest boy” “Heaviest boy heavier than heaviest girl” “Most representative half of girls generally lighter than most representative half of boys”

16. Comparing groups Boys Girls “Lightest girl same as lightest boy” “Heaviest boy same as heaviest girl” “All of the most representative half of girls lighter than most representative half of boys” “Three quarters of girls lighter than three quarters of boys”

17. Ascending height order …

18. Source: Dr Pearl’s 1938 study of 100,000 non smokers 25,000 people The Queue of DEATH! Come on guys, this is so SLOW!

19. Source: Dr Pearl’s 1938 study of 100,000 smokers 25,000 people The Queue of DEATH! Woah there! I’m not ready yet!

20. smokers non-smokers Direct comparisons easy with box plots

22. 23 boys and 11 girls were given a maths test. Their scores are listed below: Boys: 7, 13, 15, 19, 35, 35, 37, 43, 44, 44, 45, 46, 47, 47, 49, 51, 52, 55, 55, 56, 78, 82, 91 Girls: 7, 18, 23, 47, 58, 63, 68, 72, 72, 75, 87 Use box plots to compare the differences between the boys and girls scores and comment on the differences. Which scores (if any) might be considered ‘outliers’ and why (/why not)?

23. 23 boys and 11 girls were given a maths test. Their scores are listed below: Boys: 7, 13, 15, 19, 35, 35, 37, 43, 44, 44, 45, 46, 47, 47, 49, 51, 52, 55, 55, 56, 78, 82, 91 Girls: 7, 18, 23, 47, 58, 63, 68, 72, 72, 75, 87 BoysGirls Min77 LQ3523 Median4663 UQ5572 Max9187 IQR2049 Range8480 11 9 }}}} 40 9 }}}} negative skew symmetrical distribution

24. 0 10 20 30 40 50 60 70 80 90 100 (Maths score out of 100) Box plot of boys and girls maths scores B G

25. Looking for ‘outliers’ When do we feel our ‘extreme’ data is just TOO extreme?

26. Outliers High Outliers > UQ + 1.5 x IQR Low Outliers < LQ – 1.5 x IQR Eg. For boys 1.5 x IQR = 1.5 x 20 = 30 Scores less than LQ – 30 (35 – 30 = 5) are outliers Scores more than UQ + 30 (55 + 30 = 85) are outliers The only outlier is the score of 91 … but that is not such an unreasonable score! This is just a guide! For girls outliers are 145.5 … so no outliers there!

27. Plenary 1)What’s the point in Box plots? 2)Give some advantages and disadvantages of Box plots over other methods of comparing data.