Statistics For Social Sciences – MATH1208
Unit 2 – Presentation of Data
Data can be presented using a frequency table, bar graph, line graph, histogram, pie chart and a pictogram.
Frequency Table
A frequency table consists of two to three columns where the first column gives the description of the data, the second column displays the tally and the third shows the frequency of the data.
An ungrouped frequency table does not have a class interval, e.g. 0 – 5, 6 – 11, 12 – 17 and so.
1. Some students in a class counted the number of books in their bags on a certain day. The
number of books in each bag is shown below. 5 4 6 3 2 1 7 4 5 3 6 5 4 3 7 6 2 5 4 5 5 7 5 4 3 2 1 6 3 4
a) Draw a frequency table to represent the information above.
b) How many students are in the class? c) What is the modal number of books? d) How many students had:
i. 5 or more books in their bag?
iv. at least 6 books in their bag? v. less than 2 books in their bag?
e) What percentage of the students had: i. 7 books in their bag?
ii. the least number of books in their bag? iii. 4 or 5 books in their bag?
f) What is the total number of books in the class? Ans: 127 books
2. The data below shows the participants in a dart competition. The score (raw score) of each participant is listed below.
5, 7, 3, 0, 1, 4, 5, 2, 1, 3, 4, 2, 1, 5, 2, 5, 2, 0, 3, 1, 4, 0, 3, 2, 4, 2.
b. How many persons participated in the dart competition?
c. What is the modal score?
d. How many participants scored: i) more than 2?
ii) between 0 and 4? iii) at most 2?
iv) at least 4?
e. What percentage of the participants scored:
i) the maximum score? ii) 3 or 4?
A group frequency table consist of a class intervals.
Exercise
1. The data below shows the number of pens students took with them on a particular day. a) Copy and complete the grouped frequency table to represent the data given below.
7 10 6 4 11 6 1 5 2 8 6 1 2 5 10 13 10 11 4 1 2 3 1 2 1 3 4 5 6 10 11 15 14 2 6 1
Number of Pens (x) Tally Frequency
1 – 5 6 – 10 11 – 15 16 – 20
b) What is the modal class?
c) What is the total number of pens in this data?
Answer 301 pens
2. The data below shows the masses (in kg) of some students at the same school.
62 47 18 38 65 27 39 41 50 20 67 18 30 17 63 24 42 51 33 60 23 55 63 49 39
a) Copy and complete the table below.
Masses (in kg) Tally Frequency
24 – 32 33 – 41
b) What is the modal class?
c) What is the total number of students in this data?
Grouped Frequency Distribution
Consider: The masses of 60 students correct to the nearest kilograms as shown the frequency table below.
35 – 39 14
40 – 44 12
45 – 49 13
50 – 54 11
55 – 59 10
Class Interval
The class interval is a grouping of statistical
data. It enables large data to be represented in a much simpler way. The first class interval for the grouped frequency table above is 35 – 39 kg, the second is 40 – 44 kg and the fourth class interval is 50 – 54 kg.
The class limits are the end values of a class interval. Each class interval has two limits,
namely, the lower class limit (the value on the left) and the upper class limit (the value on the right).
The lower class limit for the first class interval is 35 kg
The lower class limit for the third class interval is 45 kg
The upper class limit for the second class is 44 kg
The upper class limit for the fifth class is 59 kg
Class Boundaries
35 – 39 40 – 44 45 – 49 50 – 54 55 – 59
59.5 Class
Boundaries Class Intervals
Boundaries (L.C.B.)
Intervals Boundaries
(U.C.B.)
34.5 35 – 39 39.5
39.5 40 – 44 44.5
44.5 45 – 49 49.5
49.5 50 – 54 54.5
54.5 55 – 59 59.5
Class Mid-point/Class Mark
The mid-point of a class interval is define as the average of the lower and upper class boundaries. The mid-point of a class interval is very
can also be defined as the average of the lower and upper class limits of the class interval.
The Class Mid-point or Class Mark =
The class mid-point for the 1st class interval =
The class mid-point for the 2nd class interval =
OR
The class mid-point for the 1st class interval =
The class mid-point for the 2nd class interval =
The Class Width of a Class Interval (or Class Size)
The width of a class interval (or class size) is the difference between the upper and lower class boundaries.
The width of the first class interval is U.C.B – L.C.B. = 39.5 – 34.5 = 5 kg
NOTE: The width of a class is not equal to the upper limit – the lower limit.
Exercise
1. The scores obtained by 100 children in a competition are summarized in the table below.
Score Class mid-point(x) Frequency (f) f x
0 – 9 4.5 8 36
10 – 19 14.5 13 188.5 25
22 20
50 – 59 12
Total 100
ii) class mid-points iii) class boundaries iv) class limits
b. i) State the modal class interval.
ii) State the class interval in which a score of 19.4 would lie.
2. The table below shows the distribution of
masses of 100 adults measured to the nearest kg. a) Copy and complete the table below.
Mass(kg) Frequency LL UL LCB UCB Class
Midpoint
Class Size
33 17 8
b) What is the modal class?
Rank
The rank of a set of data is the positioning of each data after arranging them in descending order.
Exercise
1. The following are some scores obtained by some students in a test. Rank the test scores for each subject.
a) Mathematics Test Scores (in %)
b) English Test Scores (in %)
87 100 55 98 96 78 87 96 100 63 82 96
The Range (Page 401, Vol. 1, R. Toolsie)
The range of a set of numbers is defined as the difference between the largest and smallest
number in the set. For the set of numbers 5, 3, 6, 4, 7, 5, the range is sometimes given by quoting the smallest and largest numbers or indicated as 3 – 7. The range of this set of numbers is
7 – 3 = 4.
Exercise
Ans: Range = $47
Cumulative Frequency Curve
There are two ways of drawing a cumulative frequency curve (or ogive). The cumulative frequency is always shown on the y-axis (vertical axis).
A cumulative frequency curve (ogive) can be drawn by plotting the cumulative frequency against the corresponding upper limit of each class interval. A smooth curve is then drawn through the points to obtain the ogive.
Another way of drawing the cumulative
is then drawn through the points to obtain the ogive.
Quartile
A quartile is one of three values that divide an ordered (ascending/descending) set of data into four equal parts. The first (or lower) quartile Q1
is the value below which one-quarter of the data lies. The lower quartile Q1 is the middle value of
the bottom half of the data.
Upper class limit Or
Upper class boundary Cumulative
frequency
0
Ogive
Q1 Q2 Q3 y
The second (or middle) quartile Q2 is the value
below which one-half of the data lies. This
quartile is known as the median. The median Q2
is the middle value of the whole data.
The third (or upper) quartile Q3 is the value
below which three-quarters of the data lies. The upper quartile Q3 is the middle value of the top
half of the data.
The inter-quartile range (I.Q.R.) = Upper
quartile (Q3) – Lower Quartile (Q1)
The semi-interquartile range (S.I.Q.R.) =
(Q3 – Q1).
For a grouped data, the lower quartile (Q1) is the
the total frequency of the data. The median (Q2)
is the value corresponding to the nth term. The
upper quartile (Q3) is the value corresponding to
the nth term.
Exercise
1. The scores obtained by 9 students in a
Mathematics test out of 20 are as follows: 5, 2, 8, 6, 1, 19, 10, 12, 3. Find the:
a. i) lower quartile (Q1) Answer 2.5
ii) upper quartile (Q3) Answer 11
iii) inter-quartile range (I.Q.R.) Answer 8.5
iv) semi-interquartile range (S.I.Q.R.)
2. Given the following heights (in cm), find the interquartile range and semi-interquartile range. a. 153, 168, 164, 151, 166, 169, 165
Answer IQR = 15 , SIQR = 7.5
b. 168, 153, 164, 151, 167, 166, 169, 165
Answer IQR = 9 , SIQR = 4.5
c. 163, 158, 154, 161, 156, 159, 155
Answer IQR = 6 , SIQR = 3
d. 158, 163, 154, 161, 157, 156, 159, 155
Answer IQR = 4.5 , SIQR = 2.25
GRAPH BOOK NEEDED
1 cm to 5 units, 1 cm to 10 units or 1 cm to 50 units and so on, for each axis.
Exercise
Answer the following.
1. The table below shows the scores obtained by some students on a mathematics test.
a. Copy and complete the table below.
Marks(x) Frequency Upper Class
Boundary
Cumulative frequency
21 – 30 11 10 14 6
b. Use a scale of 1 cm to represent 5 marks on the x-axis (horizontal axis) and 1 cm to represent 5 students on the y-axis (vertical axis). Use the upper class boundary to draw a cumulative
frequency curve to represent the data.
c. Determine the lower quartile, median and upper quartile from your graph.
d. Use the upper limit to draw a cumulative frequency curve to represent the data.
Length (x mm)
Frequen
cy UpperClass Bounda
ry
Leng th (x m
m) – Uppe r Limit Cumulativ e Frequenc y 11 – 15 2 16 – 20 4 21 – 25 8 14 6 4 2
y-axis. Use the upper class boundary to draw a cumulative frequency curve to represent the data.
c. Determine the lower quartile, median and upper quartile from your graph.
d. Use the upper limit to draw a cumulative frequency curve to represent the data.
Frequency Polygon
A frequency distribution can be represented graphically by drawing a line graph called a frequency polygon.
The frequency polygon for grouped data is
then drawing straight lines in order to join consecutive points.
Exercise
Answer the following.
1. The table below shows the scores obtained by some students on a mathematics test.
a. Copy and complete the table below.
Marks Frequency Mid-point (x)
1 – 10 2 11 – 20 7 21 – 30 11
b. Use a scale of 1 cm to represent 5 marks on the x-axis (horizontal axis) and 1 cm to represent 5 students on the y-axis (vertical axis). Draw a frequency polygon to represent the data.
2. The table below shows the length of different leaves on an apple tree.
a. Copy and complete the table below. Length
(x mm)
Frequen cy
Mid-point (x mm)
11 – 15 2
16 – 20 4
21 – 25 8
14 6 4
b. Use a scale of 2 cm to represent 10 mm on the x-axis and 2 cm to represent 10 leaves on the y-axis. Draw a frequency polygon to represent the data.
Histogram
A histogram consists of bars joined together, where the height of each bar corresponds to the frequency of the data. The histogram for a
The table below shows the frequency
distribution of the ages of some students at a tertiary institution who pursued Engineering. Copy and complete the table.
Exercise
Answer the following.
1. The table below shows the scores obtained by some students on a mathematics test.
Marks Frequency Mid-point (x)
1 – 10 2
11 – 20 7 21 – 30 11
10 14 6
b. Use a scale of 1 cm to represent 2 students on the y-axis (vertical axis). Use the mid-point to draw a histogram to represent the data above.
2. The table below shows the length of different leaves on an apple tree.
a. Copy and complete the table below.
th (x m
m) ncy class boundar ies class boundar ies 11 – 15 2 16 – 20 4 21 – 25 8 14 6 4 2
b. Use a scale of 2 cm to represent 10 mm on the x-axis and 1 cm to represent 2 leaves on the y-axis. Use the class boundaries to draw a
histogram to represent the data above.
Line graphs are useful means of displaying
statistical data to examine trends or growth over a period of time. It is used in business where
companies may predict growth in income or sales over a period of time. In medicine,
physicians use these graphs to monitor blood sugar levels, the blood pressure or even the temperature of patients.
A line graph is constructed by joining a set of points together in a consecutive manner.
Exercise
Answer the following.
1. A businessman recorded his profits over a five year period in the table below.
Profit 10000 12000 13000 10000 15000 18000
a. Represent this data on a line graph. b. Comment on any noticeable trend.
c. Which period had the largest increase in profit?
d. Which period had the lowest increase in profit?
e. What was the smallest profit observed over the five year period?
f. What do you expect to happen to the profit in the year 2012? Why?
2. The table below shows the quantity of
Year Quantity of bananas in tonnes
1989 250 1990 450 1991 200 1992 500 1993 600
a. Draw a line graph to represent this data.
b. Use your line graph to answer the following. i. During which period was there the smallest increase in the quantity of bananas?
ii. In which year was the quantity of bananas the lowest?
iv. How many bananas were produced in the year 1990?
v. During which period was there the largest increase in the quantity of bananas?
Bar Charts
A bar chart consists of a number of vertical or horizontal rectangular bars of the same width and are evenly spaced out. The height or length of each rectangular bar is directly proportional to the size of the data it is representing. Bar charts are mostly used to represent discrete data.
Types of Bar Charts:
(c) Component Bar Chart.
(a) Simple Bar Chart:
1. Simple bar chart consists of vertical or horizontal bars of equal width.
2. The length of the bars is taken
proportionately to the magnitude of the values represented. The width of the bars has no
significance.
3. If the data do not relate to time, then they should be arranged in ascending or descending order of magnitude.
Year Exports
1948 138 1951 406 1961 378 1971 683 1981 2958 1991 6168 2001 9202 2005 14410
(b) Multiple Bar Chart:
2. Grouped bars are used to represent related sets of data. For example, imports and exports of a country together are shown in multiple bar
chart.
3. Each bar in a group is shaded or coloured differently for the sake of distinction.
Years Rs. (billion)Imports Rs. (billion)Exports
(c) Component Bar Chart:
1. This chart consists of bars which are sub-divided into two or more parts.
2. The length of the bars is proportional to the totals.
Current and Development Expenditure – Pakistan (All figures in Rs. Billion)
Years ExpenditureCurrent DevelopmentExpenditure ExpenditureTotal
1988-89 153 48 201
1989-90 166 56 222
1990-91 196 65 261
1991-92 230 91 321
1992-93 272 76 348
1993-94 294 71 365
1994-95 346 82 428
(d) Percentage Component Bar Chart:
2. All the bars are of equal length showing the 100%. These bars are sub-divided into component bars in proportion to the percentages of their components.
Areas Under Crop Production (1985-90) (‘000 hectors)
Year Wheat Rice Others Total
1985-86 7403 1863 1926 11192 1986-87 7706 2066 1906 11678 1987-88 7308 1963 1612 10883 1988-89 7730 2042 1966 11738 1989-90 7759 2107 1970 11836
Percentage Areas Under Production Year Wheat Rice Others Total
Exercise
Answer the following.
2. The table below shows the quantity of
bananas (in tonnes) grown annually on a farm over the
period 1989 to 1993. Types of cars Frequency Honda 55 Toyota 70 Benz 50 BMW 35 Fiat 20
Year Quantity of bananas in tonnes
1989 250 1990 450 1991 200 1992 500
a. Draw a bar chart to represent the information given.
b. How many vehicles were in the survey?
a. Draw a bar chart to represent the information given.
b. How many bananas were grown over the period 1989 to 1993?
Pictograms
A pictograph is a chart displaying data in a pictorial form. Each picture in a graph can
represent one or more quantities, denoted by a given key. The pictograph allows us to make comparisons at a glance so that analyses can be made and decisions taken.
Answer the following.
1. The table below shows the number of smarties in a container.
a. Which colour has the highest frequency? b. What is the colour with the lowest
frequency?
c. How many more red smarties are there than blue?
e) How many green smarties are in the container?
2. The pictogram below shows how many apples were sold at a shop last week.
a. How many apples were sold on Friday? b. Which day had the least sale of apples?
d. How many less apples were sold on Monday than Saturday?
Pie Chart
A pie chart is a circular diagram, which is
another way of representing statistical data. The circle is divided into sector of varying sector
angles or areas. Each sector angle or area is directly proportional to the size of the
information it is representing.
Exercise
Answer the following.
2. Khadean spent $60 on food, $40 on transportation and $80 on phonecards.
a. How much money did she spend altogether? b. Determine the sector angle that will represent the amount spent on each quantity.
c. Construct a pie chart of radius of 5 cm, to represent the information given.
a) Determine the amount money received by:
i. Kaias ii. Tony iii. James iv. Hugo
b) What is the sector angle for:
i. Kaias ii. Tony iii. James iv. Hugo
c) Calculate the percent of the money
3. An agricultural plot of 24 hectares is divided for planting 6 hectares of potatoes, 4 hectares of peas, 10 hectares of corn and the remainder is planted in carrots.
a. Calculate the area that is to be planted in carrots.
b. Determine the sector angle that will represent the area to be planted by each crop.
