Cumulative Frequency and Box

Plots

Learning Objectives

To be able to draw a cumulative frequency curve and use it to estimate the median and interquartile range and draw a box plot

Does anybody remember how to draw a Cumulative Frequency Diagram?

Cumulative Frequency Curves

• Cumulative Frequency = a running total of the data

• Median = Half way up the Cumulative Frequency

• Lower Quartile (LQ) = ¼ way up Cum Freq

• Upper Quartile (UQ) = ¾ way up Cum Freq

• Inter Quartile Range = UQ - LQ

Big Worked ExampleThe table below shows the number of minutes students were late for their fun algebra lesson.

(a) Draw a Cumulative Frequency Diagram of the data

(b) Use it to find the Median, Lower Quartile, Upper Quartile, and Inter Quartile Range

(c) Draw a Box-Plot assuming a minimum time of o minutes and a maximum of 25 minutes

Time t (mins)Number of Students

Cumulative Frequency

0 < t ≤ 5 10

5 < t ≤ 10 16

10 < t ≤ 15 30

15 < t ≤ 20 22

20 < t ≤ 25 2

5 10 15 20 25 t mins

Cum freq

20

40

60

80

x

x

x

x x

Median = Middle Value

QUARTILES•Lower Quartile = ¼ way•Upper Quartile = ¾ way

Interquartile Range

8½ 15½12½

= 15½ - 8½ = 7 mins

Does anybody have any idea what a Box Plot is?

Box Plots

• Box Plots (Box and Whisker Diagram) are another way of displaying data for comparison

• To draw a Box Plot we need FIVE pieces of information:– 1. The Minimum Value– 2. Lower Quartile (Q1)– 3. The Median (Q2)– 4. Upper Quartile (Q3)– 5. Maximum Value

• They are drawn in the following way:

• BOX = The location of the middle 50% of the data

• WHISKERS = How the data is spread overall

3 4

12 5

10

20

30

40

50

60

70

0

Cu

mu

lati

ve F

req

uen

cy

10 20 30 40 50 60 70

Minutes Late

Med

ian =

27

LQ =

21 UQ

= 3

8

IQR = 38 – 21 = 17

mins

½

¼

¾

0 10 20 30 40 50 60

Box Plot from Cumulative Frequency Curve

Box Plot

5 10 15 20 25 t mins0

Lowest value

Upper Quartile

Highest value

Lower Quartile

Median

Work

• Ex 7:1– Page 62– Questions 5 and 6