Interpreting data … Drawing and comparing Box and Whisker diagrams (Box plots)
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Transcript of Interpreting data … Drawing and comparing Box and Whisker diagrams (Box plots)
Interpreting data …
Drawing and comparingBox and Whisker diagrams
(Box plots)
Learning objectives
All: (B grade)• Calculate quartiles and draw box and
whisker diagramsMost: (A grade)• Interpret box and whisker diagrams
and use to compare datasetsSome: (Stats GCSE)• Use all terminology and use IQR to
find “outliers”
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 thesechildren look like from the list …
• 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
Stem and leaf …
• 28, 29, 31, 35, 35, 36, 37, 41, 42, 43, 44, 48, 50, 52, 63
2 8 93 1 5 5 6 74 1 2 3 4 85 0 26 3
Key: 2 9 means 29
ORDERED STEM & LEAF
Another useful summary• A diagram to show:
min (28KG), max (63KG), median (41KG) …
Min Median Max
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
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
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
Add that to our box plot• A diagram to show:
min (28KG), lower quartile = 35KG
max (63KG), upper quartile = 48KG median (41KG) …
MinMedian
MaxLQ UQ
Some terminology
MinMedian
MaxLQ UQQ0
Q1Q2 Q3 Q4
Alternative names for quartiles
Some terminology
UQ – LQ = Interquartile Range (IQR)
Max – Min = Range
Some terminology
Positive skew: median closer to LQ than UQ
Negative skew: median closer to UQ than LQ
Symmetrical distribution
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)
Comparing groupsBoys
Girls
“Lightest girl lighter than lightest boy”
“Heaviest boy heavier than heaviest girl”
“Most representative half of girls generallylighter than most representative half of boys”
Comparing groupsBoys
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”
Ascending height order …
Source: Dr Pearl’s 1938 study of 100,000 non smokers
25,000people
25,000people
25,000people
25,000people
The Queue of DEATH!
Come on guys, this is so SLOW!
Source: Dr Pearl’s 1938 study of 100,000 smokers
25,000people
25,000people
25,000people
25,000people
The Queue of DEATH!
Woah there!I’m not
ready yet!
smokers
non-smokers
Direct comparisons easy with box plots
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, 91Girls: 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 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, 91Girls: 7, 18, 23, 47, 58, 63, 68, 72, 72, 75, 87
Boys GirlsMin 7 7LQ 35 23Median 46 63UQ 55 72Max 91 87
IQR 20 49Range 84 80
11
9
}
}40
9
}
}negative
skewsymmetricaldistribution
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
Looking for ‘outliers’
When do we feel our ‘extreme’ data isjust TOO extreme?
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 notsuch an unreasonable score! This is just a guide!
For girls outliers are < -50.5 or > 145.5 … so no outliers there!
Plenary
1) What’s the point in Box plots?2) Give some advantages and
disadvantages of Box plots over other methods of comparing data.