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§4: Cumulative frequency, drawing an ogive and other related concepts

The material in this workshop covers some of the aspects of the following Core Assessment Standards:A S 10.4.1 (a)Collect, organise and interpret univariate numerical data in order to determine measures of dispersion, including quartiles, percentiles and the interquartile rangeA S 11.4.1 (a) Calculate and represent measures of central tendency and dispersion in

univariate numerical data by drawing Ogives

OGIVES

The word ogive is used to describe various smooth curved surfaces.

In ballistics and aerodynamics, an ogive is a pointed, curved surface used to form the approximately streamlined nose of a bullet, shell, missile or aircraft.The ogive nose cone is probably one of the most common shapes used in rocketry. It exhibits very good drag characteristics.

In architecture an ogive is an arch with a pointed apex. It is formed by the intersection of two S curves usually confined to decoration. An ogive or ogival arch is a pointed, "Gothic" arch, drawn with compasses or with arcs of an ellipse

In woodworking, an ogive is a type of curve in which a piece of wood can be shaped.

In glaciology, ogives are bands of light and dark ice on the surface of a glacier.

Ogives is also a set of pieces for the piano by the composer Erik Satie.

In statistics, an ogive is the curve of a cumulative distribution function which forms an S shape. The ogive is also called a cumulative frequency curve.

Written by Jackie Scheiber and Meg Dickson© RADMASTE Centre, University of the Witwatersrand – May 2007

http://www.pitt.edu/~medart/menuglossary/ogee.htm

http://www.geocities.com/rocketguy_101/ogive/OgiveNoseCones.htm

http://www.fact-index.com/c/cu/cumulative_distribution_function_1.html

http://www.fact-index.com/s/st/statistics.html

http://www.fact-index.com/e/er/erik_satie.html

http://www.fact-index.com/p/pi/piano.html

http://www.fact-index.com/g/gl/glacier.html

http://www.fact-index.com/g/gl/glaciology.html

http://www.fact-index.com/w/wo/woodworking.html

http://www.fact-index.com/a/ar/arch.html

§4: Cumulative frequency, ogives, and percentiles FET Data Handling

FREQUENCY TABLES

In a frequency table you keep count of the number of times a data item occurs by keeping a tally. The number of times the item occurs is called the frequency of that item.

In a frequency table you can also find a ‘running total’ of frequencies. This is called the cumulative frequency. It is useful to know the running total of the frequencies as this tells you the total number of data items at different stages in the data set.

You can draw up tables showing cumulative frequencies for either grouped or ungrouped data.

1) A CUMULATIVE FREQUENCY TABLE WITH UNGROUPED DATA

Suppose you kept a record of the maths marks obtained by learners in a test in Grade 9. The test is out of a 10 marks and you have drawn a frequency table to show the marks.

A tally of the marks is shown in the table below:

Mark Tally Frequency

1 /2 ///3 ////

4

//// /

5 //// ////6 //// //// /7 //// //// ////8 //// //// //// ///9 //// ////10 ////

Total

On the next page another frequency table for this data is shown.

Note that another column labelled cumulative frequency has been added to the table.

Written by Jackie Scheiber and Meg Dickson© RADMASTE Centre, University of the Witwatersrand – April 2008

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Complete the frequency table – fill in the frequency column and the totals.


Mark Frequency

Cumulative frequency This tells you that

1 1 1 1 learner scored 1 mark2 3 3 + 1 = 4 4 learners scored marks of 2 or less3 4 4 + 4 = 8 8 learners scored marks of 3 or less

4 6 6 + 8 = 14 14 learners scored marks of 4 or less

5 9 9 + 14 = 23 23 learners scored marks of 5 or less6 11 11 + 23 = ………7 15 15 + …….…..= ……..8 18 18 + 49 = 679 10 10 + ….…….=………

10 5 5 + .……..=……….Total 82

Note: The final total in the cumulative frequency column is the same as the total

number of learners. This helps you check that your working is correct. The frequency tells you how many learners scored a certain mark

The cumulative frequency tells you how many learners scored that mark or less.

e.g.1) From the frequency column you know that 18 learners scored 8 marks.

From the cumulative frequency column you can see that 67 learners scored a mark of 8 or less

2) You can also see that 8 learners got 3 marks or less than 3 marks. Therefore, you can say that 82 – 8 = 74 learners got more than 3 marks.

Activity 11)

a) Complete the cumulative frequency table above.

b) How many learners scored 6 marks or less?

c) How many learners scored more than 6 marks?

d) How many learners scored 9 marks or less?


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2) The table below shows the number of pets owned by the 26 learners in Grade 5.

Number of pets 0 1 2 3 4 5frequency 8 6 6 3 2 1Cumulative frequency

a) Complete the table.

b) With a partner discuss i) How many learners had 3 pets or fewer than 3 pets

ii) How many learners had more than 3 pets

iii) How many learners had 4 pets or less than 4 pets

2) A CUMULATIVE FREQUENCY TABLE WITH GROUPED DATA

a) Grouped discrete data.Discrete data is data that has a certain exact value (often whole numbers) and is often collected by counting. e.g. A set of data collected about shoe sizes would be discrete data.

Discrete data can be shown by points on a number line.

Suppose you kept a record of the marks of a maths exam of the 140 learners in Grade 10. The exam is out of 100 and you have grouped the marks and recorded the data in a frequency table shown below:

Mark Frequency

Cumulative frequency This tells you that

1 - 10 0 0 0 learners scored between 1 and 10 marks

11 - 20 2 2 + 0 = 2 2 learners scored 20 marks or less 21 - 30 6 6 + 2 = 8 8 learners scored 30 marks or less31 – 40 7 7 + 8 = 15 15 learners scored 40 marks or less41 - 50 14 14 + 15 = …..51 - 60 2061 - 70 3571 - 80 2981 - 90 16

91 - 100 11


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Total 140


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Note: The frequency tells you how many learners scored between certain marks

The cumulative frequency tells you how many learners scored the upper limit or less than the upper limit of the group.

e.g.From the cumulative frequency column you can see that 15 learners scored a mark of 40 or less. Therefore, you can say that 140 – 15 = 125 learners scored more than 40 marks (i.e. they scored 41 or more marks).

b) Grouped continuous data

Continuous data can be any value within a certain range (not exact values) and is found by measuring. For example, a set of data relating to the length of people’s feet would be continuous data.

Continuous data can be shown by a continuous number line.

Suppose you measured the heights of all the learners in Grade 1. The data is shown in the following frequency table. A cumulative frequency column has been added to the frequency table.

Height, h cm Frequency Cumulative frequency This tells you that

90 < h 95 5 5 5 learners had a height of 95 cm or less

95 < h 100 9 9 + 5 = 14 14 learners had a height of 100cm or less than 100cm

100 < h 105 17 17 + 14 = 31 31 learners had a height of 105 cm or less than 105 cm

105 < h 110 28 28 + 31 = …….110 < h 115 21

115 < h 120 10Total 90

Note: The frequency for each group tells you how many learners are between the

limits of a particular group height The cumulative frequency for each group tells you how many learners are

less than or equal to the upper boundary of the group.

e.g.1. From the frequency column you know that 17 learners are between 100 and

105 cm tall (100 < h 105).From the cumulative frequency column you can see that 31 learners are 105 cm or less than 105 cm tall

2. From the cumulative frequency column you know that 14 learners are 100 cm or less than 100 cm. Therefore, you can also tell that 90 – 14 = 76 learners are taller than 100 cm.


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Activity 21)

a) Complete the cumulative frequency table on the previous pageb) How many learners were between 105 cm and 110 cm in height?

c) How many learners were 110 cm or less than 110 cm in height?

d) How many learners were greater than 110 cm in height?

2) The table shows the marks for a maths exam in Grade 10

% Number of learners

Cumulative frequency

0 < x 10 010 < x 20 220 < x 30 630 < x 40 740 < x 50 1450 < x 60 2060 < x 70 3570 < x 80 2980 < x 90 16

90 < x 100 11Total = 140

a) What do you notice that is different in this cumulative frequency table compared to the one on page 4?

b) Complete the table by filling in the cumulative frequency column.

c) Your HOD tells you that any learner that got 40% or less must re-write the exam. How many learners must re-write the exam.

d) If a learner scored more than 80% he/she achieved an A symbol. Discuss how you can work out how many learners got As.

e) How would you be able to find out the number of learners who scored 80% or higher than 80%?


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REPRESENTING DATA FROM A FREQUENCY TABLE.

This frequency table shows Grade 12 maths exam marks

% Number of learners

0 < x 10 010 < x 20 220 < x 30 630 < x 40 740 < x 50 1450 < x 60 2060 < x 70 3570 < x 80 2980 < x 90 6

90 < x 100 1

total 120

This data can be represented by drawing a histogram.


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This same information can be represented in a frequency polygon by joining the midpoints of the bars of the histogram like this

and then erasing the bars like this:

Notice: the graph begins and ends where the frequency is zero

A frequency polygon is a line graph of the data. It is a useful way of representing the data as it gives another way of reading frequency. Frequency polygons are often used to compare different frequency distributions. You may have seen line graphs in the newspaper that are actually frequency polygons.


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DRAWING A CUMULATIVE FREQUENCY CURVE OR OGIVE

Another way of representing the data from the frequency table is to draw a cumulative frequency curve. When you draw the graph of a cumulative frequency distribution you obtain a curve that has a characteristic shape. This curve is called a cumulative frequency curve or an ogive.

Note: The frequency polygon and the cumulative frequency curve (or ogive) are

very different in shape even though they represent the same set of data. The frequency polygon follows the shape of the histogram and shows the

frequency of each group interval of the marks. The ogive shows the running total of the frequencies and the marks. The ogive has a distinct S shape. The ogive begins where the frequency (but not necessarily the marks) is zero.

1) DRAWING AN OGIVE OF UNGROUPED DISCRETE DATA.The following activity will take you through how to draw a cumulative frequency curve for ungrouped discrete data


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Activity 3The cumulative frequency table of the Grade 9 maths test given on page 2 is shown below. You are going to draw a cumulative frequency curve to show the information. To do so you need ordered pairs to plot on the Cartesian plane. The first element of the ordered pair is the mark and the second element of the ordered pair is the cumulative frequency value associated with that mark.

1) Complete the table

Mark Frequency

Cumulative frequency points

0 0 0 (0 ; 0)1 1 1 (1 ; 1)2 3 4 (2 ; 4)3 4 8 (3 ; 8)4 6 145 9 236 11 347 15 498 18 679 10 77

10 5 82total 82

2) Plot the points from the frequency table on to the grid below. Join the points with a smooth line.


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Note: If you join the points with a smooth line you get a cumulative frequency curve. The graph starts on the horizontal axis – i.e. where the cumulative frequency

is zero. i.e start the curve at (0 ; 0) The shape of the curve is a leaning S shape. If you get any other shape you

have made a mistake.

2) DRAWING AN OGIVE OF GROUPED CONTINUOUS DATAFor a cumulative frequency curve of continuous data, the first element of the ordered pair is the upper limit of the interval. the second element is the value of the cumulative frequency.

The cumulative frequency table of the heights of the learners in Grade 1 given on page 5 is shown below:

Height, h cm Frequency Cumulative frequency Points

85 < h ≤ 90 0 0 (90 ; 0)90 < h 95 5 5 (95 ; 5)

95 < h 100 9 14 (100 ; 14)100 < h

105 17 31

105 < h 110 28 59

110 < h 115 21 80

115 < h 120 10 90

Total = 90

The graph looks like this:


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heights of learners in grade 1

05

101520253035404550556065707580859095

90 100 110 120

height (cm)

cum

ulat

ive

frequ

ency

Fill in the rest of the points


The frequency curve allows you to read off the numbers of learners who are less than a certain height. If you wanted to know how many learners were 102 cm or were shorter than 102 cm you can read up from 102 cm on the horizontal axis and find that there are about 20 learners who are shorter than or equal to 102cm - see arrows on graph.


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Note: The graph starts at the point where the cumulative frequency is zero (at the

lower end of the first group. So you plot the point (90 ; 0) ). When you join the points on the graph you are assuming that the items are

evenly spread throughout the groups. Data items are not necessarily evenly spread on a graph so any reading from a frequency curve is an estimate, not an exact reading.

Always draw lines on the graph to show where you took the readings

Activity 41) Use the graph on the previous page:

a) Estimate how many learners were 110 cm or less in height. Draw lines on the graph to show where your answer came from.

b) Does this correspond to the number on the cumulative frequency table?

c) How many learners were more than 110 cm in height?

2) Forty Five Grade 10 learners were asked to estimate how long they thought one minute is. They indicated when they thought one minute was up, and their partner recorded the length of time, correct to the nearest second. Their estimates are given below:

53 47 77 63 59 54 62 65 71 77 42 58 67 51 72 57 73 48 61 46 51 50 63 68 54 50 65 53 56 6944 56 77 58 55 79 56 58 67 52 48 70 49 71 73

a) Use the following table to work out the frequency.

Time, x, (in

seconds)Tallies Frequenc

yCumulative frequency Points

40 < x ≤ 4545 < x ≤ 5050 < x ≤ 5555 < x ≤ 6060 < x ≤ 6565 < x ≤ 7070 < x ≤ 7575 < x ≤ 80

b) On the squared paper on the next page construct a frequency polygon to illustrate the above data.

c) Complete the cumulative frequency column and find the points that have to be plotted.

d) Construct an ogive to show the data.e) Use the ogive to answer the following two questions:

i) How many estimates were 60 seconds or less?

ii) How many estimates were more than 60 seconds?

f) Write down 3 things that will help you remember the differences between a frequency polygon and a cumulative frequency curve


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THE MEDIAN AND QUARTILES FROM A CUMULATIVE FREQUENCY TABLE

Look at the cumulative frequency table of the Grade 9 maths marks on page 2. If you were to write all the data items out in ascending order you could find the median and the quartiles of the data set.

If you divide the marks of 82 learners into four groups containing the same number of marks, the divisions might look like this:

We work out the position of Q1, M and Q3 is as follows: There are 82 data items. 82 ÷ 2 = 41. So the median lies between

the 41st and 42nd item and is the average of the 41st and 42nd item.. There are 41data items to the left of the median. 41 ÷ 2 = 20 ½, so

Q1 is the score of the 21st item.

This means that we have

20 terms Q1 20 terms M 20 terms Q3 20 terms

To find Q3 we can go: 82 – 20 = 62. So Q3 is the score of the 62nd item

These values can also be found in the cumulative frequency table by counting the data items like this:

Mark Frequency

Cumulative frequency

1 1 12 3 43 4 84 6 14

5 9

23

6 11 347 15 498 18 67


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The 41st and 42nd learners are in here.Median = 7

The 21st learner is in here. Q 1= 5

Scores of learners 1 - 20



Scores of learners 63 - 825 8

Median is the average score of 41st and 42nd learners’ scores

Lower quartile is the score of the 21st learner Upper quartile is the score

of the 62nd learner

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9 1077

10 5 82Total =

82

NB: 1. The median and quartiles are the marks, not the frequencies2. This method of finding the median can be used both for discrete and

continuous data.


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The 62nd learner is in here. Q 3 = 8


THE MEDIAN AND QUARTILES FROM AN OGIVE

A cumulative frequency graph shows data in order from the smallest to the largest. The ogive of the data in the frequency table on the previous looks like this:

NB: Each point on the cumulative frequency curve gives an estimate of a value in

a particular position in the whole data set.


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Grade 10 maths marks

02468

10121416182022242628303234363840424446485052545658606264666870727476788082848688

1 2 3 4 5 6 7 8 9 10

marks

cum

ulat

ive

frequ

ncy

Median is the 41½ th value

Q 1 is the 21st value

Estimate of the median is read here.Median 7

Estimate of the lower quartile is read here.Q1 5

Q 3 is the 62nd value

Estimate of the upper quartile is read here.Q3 8


PERCENTILES

The median and the quartiles divide a data set into 4 equal parts – the same number of items in each part.

Two other measures are sometimes used when describing a data set: Deciles: They divide the data set into 10 equal parts Percentiles : They divide the data set into 100 equal parts

It follows then that The median is the 50th percentile. This means 50% of the data items are

below the median Q1 = 25th percentile. This means 25% of the data items are below Q1 Q 3 = 75th percentile. This means 75% of the data items are below Q3

Similarly The 33rd percentile means that 1/3 of the data items are below this value, The 10th percentile is the value that separates the bottom 10% of values in a

data set from the rest and the 90th percentile is the value that is 90% of the way through the data set.

Percentiles should only be used with large sets of data.

Where you don’t have exactly 100 pieces of data, you have to find the percentage of the total in order to find a particular percentile.

ExampleThe 16th percentile of the data on the previous page is found like this: 16% of 82 = 13,12 On vertical axis find 13 then read across to curve and then down to

horizontal axis 16th percentile 4 This means 16% of the class scored 4 marks or less.

The next activity gives you more practice on using the ogive to find quartiles and percentiles.


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Activity 51)

a) Use the ogive on page 15 to find an estimate of the:i) 10th percentile of the data

ii) 90th percentile

b) Write a sentence that interprets these two values

c) Complete the following sentence”

50% of the class got …………….or less out of 10 for the test.

2) Which percentile is the same as the a) Median b) Lower quartile c) Upper quartile

3) The maths marks of 100 learners are shown in the frequency table

marks frequency Cumulative frequency Points

0-10 1 1 (10 ; 1)11-20 2 321-30 1331-40 2441-50 3251-60 1661-70 1171-80 1 100

a) Complete the table.b) Plot the points on the grid on the next page to form a cumulative

frequency curve.c) Calculate the median

d) On the graph show where you might read the estimate of the median. e) Find the lower quartile and the upper quartile from the frequency table.

f) Use arrows on the curve to show where to estimate the upper and lower quartiles from the ogive. Do your values agree?


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g) Find the10th percentile. Use arrows on the curve to show where to take the readings.

h) Find the 60th percentile. Use arrows on the curve to show where to take the readings.


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Summary of the ogive.The cumulative frequency curve has links to the 5 number summary and the box-and-whisker plot. Look at the following diagram that shows this:


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0

10

20

30

40

50

60

70

80

90

100

data items

cu mu

lati

ve

fre

qu en cy

Lower quartile

Median

Upper quartile

Minimum value Maximum value

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