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Median, Quartiles, Inter-Quartile Range and Box Plots.Measures of SpreadRemember: The range is the measure of spread that goes with the mean.

Mean = 7 + 5 + 2 + 7 + 6 + 12 + 10 + 4 + 8 + 9 10 = 70

10= 7

Range = 12 – 2 = 10

Example 1. Two dice were thrown 10 times and their scores were added together and recorded. Find the mean and range for this data.

7, 5, 2, 7, 6, 12, 10, 4, 8, 9

Median, Quartiles, Inter-Quartile Range and Box Plots.Measures of SpreadThe range is not a good measure of spread because one extreme, (very high or very low value) can have a big affect. The measure of spread that goes with the median is called the inter-quartile range and is generally a better measure of spread because it is not affected by extreme values.

A reminder about the median

Single middle value

Averages (The Median)The median is the middle value of a set of data once the data has been ordered.Example 1. Robert hit 11 balls at Grimsby driving range. The recorded distances of his drives, measured in yards, are given below. Find the median distance for his drives.85, 125, 130, 65, 100, 70, 75, 50, 140, 95,

70

Median drive = 85 yards

50, 65, 70, 70, 75, 85, 95, 100, 125, 130, 140

Ordered data

Two middle values so take the mean.

Averages (The Median)The median is the middle value of a set of data once the data has been ordered.

Example 1. Robert hit 12 balls at Grimsby driving range. The recorded distances of his drives, measured in yards, are given below. Find the median distance for his drives.85, 125, 130, 65, 100, 70, 75, 50, 140, 135, 95,

70

Median drive = 90 yards

50, 65, 70, 70, 75, 85, 95, 100, 125, 130, 135, 140

Ordered data

Finding the median, quartiles and inter-quartile range.

12, 6, 4, 9, 8, 4, 9, 8, 5, 9, 8, 10

4, 4, 5, 6, 8, 8, 8, 9, 9, 9, 10, 12

Order the data

Inter-Quartile Range = 9 - 5½ = 3½

Example 1: Find the median and quartiles for the data below.

Lower Quartile = 5½

Q1

Upper Quartile = 9

Q3

Median = 8

Q2

Upper Quartile = 10

Q3

Lower Quartile = 4

Q1

Median = 8

Q2

3, 4, 4, 6, 8, 8, 8, 9, 10, 10, 15,

Finding the median, quartiles and inter-quartile range.

6, 3, 9, 8, 4, 10, 8, 4, 15, 8, 10 Order the data

Inter-Quartile Range = 10 - 4 = 6

Example 2: Find the median and quartiles for the data below.

2, 5, 6, 6, 7, 8, 8, 8, 9, 9, 10, 15Median = 8 hours and the inter-quartile range = 9 – 6 = 3 hours.

Battery Life: The life of 12 batteries recorded in hours is:2, 5, 6, 6, 7, 8, 8, 8, 9, 9, 10, 15

Mean = 93/12 = 7.75 hours and the range = 15 – 2 = 13 hours.

Discuss the calculations below.

The averages are similar but the measures of spread are significantly different since the extreme values of 2 and 15 are not included in the inter-quartile range.

4 5 6 7 8 9 10 11 12

MedianLower

QuartileUpper

QuartileLowest Value

Highest Value

Box WhiskerWhisker

130 140 150 160 170 180 190

Boys

Girlscm

Box and Whisker Diagrams. Box plots are useful for comparing two or more sets of data like that shown below for heights of boys and girls in a class.

Anatomy of a Box and Whisker Diagram.

Box Plots

Lower Quartile = 5½

Q1

Upper Quartile = 9

Q3

Median = 8

Q2

4 5 6 7 8 9 10 11 12

4, 4, 5, 6, 8, 8, 8, 9, 9, 9, 10, 12

Example 1: Draw a Box plot for the data belowDrawing a Box Plot.

Upper Quartile = 10

Q3

Lower Quartile = 4

Q1

Median = 8

Q2

3, 4, 4, 6, 8, 8, 8, 9, 10, 10, 15,

Example 2: Draw a Box plot for the data below

Drawing a Box Plot.

3 4 5 6 7 8 9 10 11 12 13 14 15

Upper Quartile = 180

Qu

Lower Quartile = 158

QL

Median = 171

Q2

Question: Stuart recorded the heights in cm of boys in his class as shown below. Draw a box plot for this data.

Drawing a Box Plot.

137, 148, 155, 158, 165, 166, 166, 171, 171, 173, 175, 180, 184, 186, 186

130 140 150 160 170 180 190cm

2. The boys are taller on average.

Question: Gemma recorded the heights in cm of girls in the same class and constructed a box plot from the data. The box plots for both boys and girls are shown below. Use the box plots to choose some correct statements comparing heights of boys and girls in the class. Justify your answers.

Drawing a Box Plot.

130 140 150 160 170 180 190

Boys

Girls

cm

1. The girls are taller on average.

3. The girls show less variability in height.

4. The boys show less variability in height.

5. The smallest person is a girl.

6. The tallest person is a boy.

10

20

30

40

50

60

70

0

Cum

ulat

ive

Freq

uenc

y

10 20 30 40 50 60 70Minutes Late

Med

ian

= 27LQ

=

21 UQ =

38

IQR = 38 – 21 = 17

mins

½

¼

¾

0 10 20 30 40 50 60

Box Plot from Cumulative Frequency Curve

CFC

? ?

We can now construct a

partial box plot from our earlier

work on cumulative frequency

curves.

Finding the median, quartiles and inter-quartile range.

12, 6, 4, 9, 8, 4, 9, 8, 5, 9, 8, 10

Example 1: Find the median and quartiles for the data below.

Finding the median, quartiles and inter-quartile range.

6, 3, 9, 8, 4, 10, 8, 4, 15, 8, 10

Example 2: Find the median and quartiles for the data below.

Worksheet 1

worksheet

Worksheet 2

3

130 140 150 160 170 180 190cm

4 5 6 7 8 9 10 11 12

3 4 5 6 7 8 9 10 11 12 13 14 15

0 10 20 30 40 50 60

1

2

4

Box Plotsworksheet

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