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

2. Learning objectives All:
Calculate quartiles and draw box and whisker diagrams
Most:
Interpret box and whisker diagrams and use to compare datasets
Some:
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. The weights (KG) of 15 children: 28, 29, 31, 35, 35, 36, 37, 41, 42, 43, 44, 48, 50, 52, 63
Minimum = 28KG
Maximum = 63KG
Range = 35KG
Mode = 35KG
Median = 41KG
Mean = 40.9KG
Slightly easier to understand with these summary figures …

5. Stem and leaf …
28, 29, 31, 35, 35, 36, 37, 41, 42, 43, 44, 48, 50, 52,
Key: 2 9 means 29
ORDERED STEM & LEAF
We now have the raw data AND a pictorial representation …

6. Stem and leaf = Bar chart?

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

8. 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) = 8th item

9. 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) = 4th item

10. 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) = 12th item

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

12. Some terminology Q2 Q1 Q3 Q0 Q4 Median LQ UQ Min Max
Alternative names for quartiles

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

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

15. 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)

16. 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”

17. 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”

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

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

20. Direct comparisons easy with box plots
non-smokers
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, , 46, 47, 47, 49, 51, 52, 55, 55, 56, , 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. symmetrical distribution
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, , 55, 55, 56, 78, 82, 91
Girls: 7, 18, 23, 47, 58, 63, 68, 72, 72, 75, 87
Boys Girls
Min LQ
Median UQ
Max
IQR
Range } 11 9
symmetrical distribution } 40 9
negative skew

24. Box plot of boys and girls maths scores
(Maths score out of 100)

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 ( = 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 < or > … so no outliers there!

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