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

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”

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 …

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 …

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 We now have the raw data AND a pictorial representation …

Stem and leaf = Bar chart?

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 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

Some terminology Q2 Q1 Q3 Q0 Q4 Median LQ UQ Min Max 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 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”

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”

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

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

Direct comparisons easy with box plots non-smokers 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, 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)?

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, 52, 55, 55, 56, 78, 82, 91 Girls: 7, 18, 23, 47, 58, 63, 68, 72, 72, 75, 87 Boys Girls Min 7 7 LQ 35 23 Median 46 63 UQ 55 72 Max 91 87 IQR 20 49 Range 84 80 } 11 9 symmetrical distribution } 40 9 negative skew

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

Looking for ‘outliers’ When do we feel our ‘extreme’ data is just 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 not such an unreasonable score! This is just a guide! For girls outliers are < -50.5 or > 145.5 … so no outliers there!

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