Cumulative Frequency Curves

Outcomes…• Calculate the cumulative frequency • Write down the upper class

boundaries• Plot the cumulative frequency curve• Find the median value• Find the upper and lower quartiles• Find the inter-quartile range

Cumulative Frequency

Height (mm)

Frequency F

40X45 45X50

7 12

50X55

15

55X60

3

Height (mm)

Frequency F

40X45 45X50

7 12

50X55

15

55X60

3

Cumulative Frequency

Height (mm)

Frequency F

40X45 45X50

7 12

50X55

15

55X60

3

Cumulative Frequency 7

Height (mm)

Frequency F

40X45 45X50

7 12

50X55

15

55X60

3

Cumulative Frequency 7 19

Height (mm)

Frequency F

40X45 45X50

7 12

50X55

15

55X60

3

Cumulative Frequency 7 19 34

Height (mm)

Frequency F

40X45 45X50

7 12

50X55

15

55X60

3

Cumulative Frequency 7 19 34 37

Upper Class Boundary

Height (mm)

Frequency F

40X45 45X50

7 12

50X55

15

55X60

3

Cumulative Frequency 7 19 34 37

Upper class boundary

Height (mm)

Frequency F

40X45 45X50

7 12

50X55

15

55X60

3

Cumulative Frequency 7 19 34 37

Upper class boundary 45

Height (mm)

Frequency F

40X45 45X50

7 12

50X55

15

55X60

3

Cumulative Frequency 7 19 34 37

Upper class boundary 45 50

Height (mm)

Frequency F

40X45 45X50

7 12

50X55

15

55X60

3

Cumulative Frequency 7 19 34 37

Upper class boundary 45 50 55

Height (mm)

Frequency F

40X45 45X50

7 12

50X55

15

55X60

3

Cumulative Frequency 7 19 34 37

Upper class boundary 45 50 55 60

Plot the curve

Cumulative Frequency 7 19 34 37

Upper class boundary 45 50 55 60

Cress seedling height

0

10

20

30

40

35 40 45 50 55 60

Height (mm)

Cum

ulat

ive

freq

uenc

y

Cress seedling height

0

10

20

30

40

35 40 45 50 55 60

Height (mm)

Cum

ulat

ive

freq

uenc

y

Cress seedling height

0

10

20

30

40

35 40 45 50 55 60

Height (mm)

Cum

ulat

ive

freq

uenc

y

Cress seedling height

0

10

20

30

40

35 40 45 50 55 60

Height (mm)

Cum

ulat

ive

freq

uenc

y

Cress seedling height

0

10

20

30

40

35 40 45 50 55 60

Height (mm)

Cum

ulat

ive

freq

uenc

y

Cress seedling height

0

10

20

30

40

35 40 45 50 55 60

Height (mm)

Cum

ulat

ive

freq

uenc

y

Cress seedling height

0

10

20

30

40

35 40 45 50 55 60

Height (mm)

Cum

ulat

ive

freq

uenc

y

Cress seedling height

0

10

20

30

40

35 40 45 50 55 60

Height (mm)

Cum

ulat

ive

freq

uenc

y

Median

Cumulative Frequency 7 19 34 37

Upper class boundary 45 50 55 60

Cress seedling height

0

10

20

30

40

35 40 45 50 55 60

Height (mm)

Cum

ulat

ive

freq

uenc

y

Cress seedling height

0

10

20

30

40

35 40 45 50 55 60

Height (mm)

Cum

ulat

ive

freq

uenc

y

Cress seedling height

0

10

20

30

40

35 40 45 50 55 60

Height (mm)

Cum

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ive

freq

uenc

y

Cress seedling height

0

10

20

30

40

35 40 45 50 55 60

Height (mm)

Cum

ulat

ive

freq

uenc

y

Median ≈ 49mm

Cress seedling height

0

10

20

30

40

35 40 45 50 55 60

Height (mm)

Cum

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freq

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y

Median ≈ 49mm

Quartiles

The lower quartile tells us what value 25% of the sample

are less than

The upper quartile tells us what value 75% of the sample

are less than

37 cress seeds in total

A quarter of 37 is 9.25Three quarters is 27.75

We use the graph to find what lengths correspond to

cumulative frequencies of 9.25 and 27.75

Cress seedling height

0

10

20

30

40

35 40 45 50 55 60

Height (mm)

Cum

ulat

ive

freq

uenc

y

Cress seedling height

0

10

20

30

40

35 40 45 50 55 60

Height (mm)

Cum

ulat

ive

freq

uenc

y

Cress seedling height

0

10

20

30

40

35 40 45 50 55 60

Height (mm)

Cum

ulat

ive

freq

uenc

y

Cress seedling height

0

10

20

30

40

35 40 45 50 55 60

Height (mm)

Cum

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ive

freq

uenc

y

LQ ≈ 45mm

Cress seedling height

0

10

20

30

40

35 40 45 50 55 60

Height (mm)

Cum

ulat

ive

freq

uenc

y

Cress seedling height

0

10

20

30

40

35 40 45 50 55 60

Height (mm)

Cum

ulat

ive

freq

uenc

y

Cress seedling height

0

10

20

30

40

35 40 45 50 55 60

Height (mm)

Cum

ulat

ive

freq

uenc

y

Cress seedling height

0

10

20

30

40

35 40 45 50 55 60

Height (mm)

Cum

ulat

ive

freq

uenc

y

UQ ≈ 52mm

Inter-quartile range

The difference between the upper and lower quartile

Median ≈ 49mm

LQ ≈ 45mm

UQ ≈ 52mm

IQR ≈ UQ – LQ ≈ 7mm

Tells you the range of values occupied by the middle 50% of

your sample

Summary

Cumulative frequency• Running total of the frequencies• Plot against the upper class boundaries• Draw a smooth curve thru’ the points• Take half the total frequency and use

that to read off a value of the median• Find the Lower Quartile and the Upper

Quartile• Find the Inter-quartile range by

subtracting

Your turn