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Introduction

Information in the form of numbers, graphs and tables is all around us; on television, on the radio or in the newspaper. We are exposed to crime rates, sports results, rainfall, government spending, rate of HIV/AIDS infection, population growth and economic growth.

This chapter demonstrates how Mathematics can be used to manipulate data, to represent or misrepresent trends and patterns and to provide solutions that are directly applicable to the world around us.

Skills relating to the collection, organisation, display, analysis and interpretation of information that were introduced in earlier grades are developed further.

Recap of Earlier Work

The collection of data has been introduced in earlier grades as a method of obtaining answers to questions about the world around us. This work will be briefly reviewed.

Data and Data Collection

Data

Definition 1: Data

Data refers to the pieces of information that have been observed and recorded, from an experiment or a survey. There are two types of data: primary and secondary. The word "data" is the plural of the word "datum", and therefore one should say, "the data are" and not "the data is".

Data can be classified as primary or secondary, and primary or secondary data can be classified as qualitative or quantitative. Figure 1 summarises the classifications of data.

Figure 1: Classes of data.
Figure 1 (MG10C16_001.png)
  • Primary data: describes the original data that have been collected. This type of data is also known as raw data. Often the primary data set is very large and is therefore summarised or processed to extract meaningful information.
  • Qualitative data: is information that cannot be written as numbers, for example, if you were collecting data from people on how they feel or what their favourite colour is.
  • Quantitative data: is information that can be written as numbers, for example, if you were collecting data from people on their height or weight.
  • Secondary data: is primary data that has been summarised or processed, for example, the set of colours that people gave as favourite colours would be secondary data because it is a summary of responses.

Transforming primary data into secondary data through analysis, grouping or organisation into secondary data is the process of generating information.

Purpose of Collecting Primary Data

Data is collected to provide answers that help with understanding a particular situation. Here are examples to illustrate some real world data collections scenarios in the categories of qualitative and quantitative data.

Qualitative Data

  • The local government might want to know how many residents have electricity and might ask the question: "Does your home have a safe, independent supply of electricity?"
  • A supermarket manager might ask the question: “What flavours of soft drink should be stocked in my supermarket?" The question asked of customers might be “What is your favourite soft drink?” Based on the customers' responses, the manager can make an informed decision as to what soft drinks to stock.
  • A company manufacturing medicines might ask “How effective is our pill at relieving a headache?” The question asked of people using the pill for a headache might be: “Does taking the pill relieve your headache?” Based on responses, the company learns how effective their product is.
  • A motor car company might want to improve their customer service, and might ask their customers: “How can we improve our customer service?”

Quantitative Data

  • A cell phone manufacturing company might collect data about how often people buy new cell phones and what factors affect their choice, so that the cell phone company can focus on those features that would make their product more attractive to buyers.
  • A town councillor might want to know how many accidents have occurred at a particular intersection, to decide whether a robot should be installed. The councillor would visit the local police station to research their records to collect the appropriate data.
  • A supermarket manager might ask the question: “What flavours of soft drink should be stocked in my supermarket?" The question asked of customers might be “What is your favourite soft drink?” Based on the customers' responses, the manager can make an informed decision as to what soft drinks to stock.

However, it is important to note that different questions reveal different features of a situation, and that this affects the ability to understand the situation. For example, if the first question in the list was re-phrased to be: "Does your home have electricity?" then if you answered yes, but you were getting your electricity from a neighbour, then this would give the wrong impression that you did not need an independent supply of electricity.

Methods of Data Collection

The method of collecting the data must be appropriate to the question being asked. Some examples of data collecting methods are:

  1. Questionnaires, surveys and interviews
  2. Experiments
  3. Other sources (friends, family, newspapers, books, magazines and the Internet)

The most important aspect of each method of data collecting is to clearly formulate the question that is to be answered. The details of the data collection should therefore be structured to take your question into account.

For example, questionnaires, interviews or surveys would be most appropriate for the list of questions in "Purpose of Collecting Primary Data".

Samples and Populations

Before the data collecting starts, it is important to decide how much data is needed to make sure that the results give an accurate reflection to the required answers. Ideally, the study should be designed to maximise the amount of information collected while minimising the effort. The concepts of populations and samples is vital to minimising effort.

The following terms should be familiar:

  • Population: describes the entire group under consideration in a study. For example, if you wanted to know how many learners in your school got the flu each winter, then your population would be all the learners in your school.
  • Sample: describes a group chosen to represent the population under consideration in a study. For example, for the survey on winter flu, you might select a sample of learners, maybe one from each class.
  • Random sample: describes a sample chosen from a population in such a way that each member of the population has an equal chance of being chosen.

Figure 2
Figure 2 (MG10C16_002.png)

Choosing a representative sample is crucial to obtaining results that are unbiased. For example, if we wanted to determine whether peer pressure affects the decision to start smoking, then the results would be different if only boys were interviewed, compared to if only girls were interviewed, compared to both boys and girls being interviewed.

Therefore questions like: "How many interviews are needed?" and "How do I select the candidates for the interviews?" must be asked during the design stage of the sampling process.

The most accurate results are obtained if the entire population is sampled for the survey, but this is expensive and time-consuming. The next best method is to randomly select a sample of subjects for the interviews. This means that whatever the method used to select subjects for the interviews, each subject has an equal chance of being selected. There are various methods of doing this for example, names can be picked out of a hat or can be selected by using a random number generator. Most modern scientific calculators have a random number generator or you can find one on a spreadsheet program on a computer.

So, if you had a total population of 1 000 learners in your school and you randomly selected 100, then that would be the sample that is used to conduct your survey.

Example Data Sets

The remainder of this chapter deals with the mathematical details that are required to analyse the data collected.

The following are some example sets of data which can be used to apply the methods that are being explained.

Data Set 1: Tossing a Coin

A fair coin was tossed 100 times and the values on the top face were recorded. The data are recorded in "Data Set 1: Tossing a coin".

Table 1: Results of 100 tosses of a fair coin. H means that the coin landed heads-up and T means that the coin landed tails-up.
H T T H H T H H H H
H H H H T H H T T T
T T H T T H T H T H
H H T T H T T H T T
T H H H T T H T T H
H T T T T H T T H H
T T H T T H T T H T
H T T H T T T T H T
T H T T H H H T H T
T T T H H T T T H T

Data Set 2: Casting a die

A fair die was cast 100 times and the values on the top face were recorded. The data are recorded in "Data Set 2: Casting a die".

Table 2: Results of 200 casts of a fair die.
3 5 3 6 2 6 6 5 5 6 6 4 2 1 5 3 2 4 5 4
1 4 3 2 6 6 4 6 2 6 5 1 5 1 2 4 4 2 4 4
4 2 6 4 5 4 3 5 5 4 6 1 1 4 6 6 4 5 3 5
2 6 3 2 4 5 3 2 2 6 3 4 3 2 6 4 5 2 1 5
5 4 1 3 1 3 5 1 3 6 5 3 4 3 4 5 1 2 1 2
1 3 2 3 6 3 1 6 3 6 6 1 4 5 2 2 6 3 5 3
1 1 6 4 5 1 6 5 3 2 6 2 3 2 5 6 3 5 5 6
2 6 6 3 5 4 1 4 5 1 4 1 3 4 3 6 2 4 3 6
6 1 1 2 4 5 2 5 3 4 3 4 5 3 3 3 1 1 4 3
5 2 1 4 2 5 2 2 1 5 4 5 1 5 3 2 2 5 1 1

Data Set 3: Mass of a Loaf of Bread

There are regulations in South Africa related to bread production to protect consumers. Here is an excerpt from a report about the legislation:

"The Trade Metrology Act requires that if a loaf of bread is not labelled, it must weigh 800g, with the leeway of five percent under or 10 percent over. However, an average of 10 loaves must be an exact match to the mass stipulated. - Sunday Tribune of 10 October 2004 on page 10"

We can use measurements to test if consumers getting value for money. An unlabelled loaf of bread should weigh 800g. The masses of 10 different loaves of bread were measured at a store for 1 week. The data are shown in Table 3.

Table 3: Masses (in g) of 10 different loaves of bread, from the same manufacturer, measured at the same store over a period of 1 week.
Monday Tuesday Wednesday Thursday Friday Saturday Sunday
802.39 787.78 815.74 807.41 801.48 786.59 799.01
796.76 798.93 809.68 798.72 818.26 789.08 805.99
802.50 793.63 785.37 809.30 787.65 801.45 799.35
819.59 812.62 809.05 791.13 805.28 817.76 801.01
801.21 795.86 795.21 820.39 806.64 819.54 796.67
789.00 796.33 787.87 799.84 789.45 802.05 802.20
788.99 797.72 776.71 790.69 803.16 801.24 807.32
808.80 780.38 812.61 801.82 784.68 792.19 809.80
802.37 790.83 792.43 789.24 815.63 799.35 791.23
796.20 817.57 799.05 825.96 807.89 806.65 780.23

Data Set 4: Global Temperature

The mean global temperature from 1861 to 1996 is listed in Table 4. The data, obtained from http://www.cgd.ucar.edu/stats/Data/Climate/, was converted to mean temperature in degrees Celsius.

Table 4: Global temperature changes over the past 135 years. There has been a lot of discussion regarding changing weather patterns and a possible link to pollution and greenhouse gasses.
Year Temperature Year Temperature Year Temperature Year Temperature
1861 12.66 1901 12.871 1941 13.152 1981 13.228
1862 12.58 1902 12.726 1942 13.147 1982 13.145
1863 12.799 1903 12.647 1943 13.156 1983 13.332
1864 12.619 1904 12.601 1944 13.31 1984 13.107
1865 12.825 1905 12.719 1945 13.153 1985 13.09
1866 12.881 1906 12.79 1946 13.015 1986 13.183
1867 12.781 1907 12.594 1947 13.006 1987 13.323
1868 12.853 1908 12.575 1948 13.015 1988 13.34
1869 12.787 1909 12.596 1949 13.005 1989 13.269
1870 12.752 1910 12.635 1950 12.898 1990 13.437
1871 12.733 1911 12.611 1951 13.044 1991 13.385
1872 12.857 1912 12.678 1952 13.113 1992 13.237
1873 12.802 1913 12.671 1953 13.192 1993 13.28
1874 12.68 1914 12.85 1954 12.944 1994 13.355
1875 12.669 1915 12.962 1955 12.935 1995 13.483
1876 12.687 1916 12.727 1956 12.836 1996 13.314
1877 12.957 1917 12.584 1957 13.139
1878 13.092 1918 12.7 1958 13.208
1879 12.796 1919 12.792 1959 13.133
1880 12.811 1920 12.857 1960 13.094
1881 12.845 1921 12.902 1961 13.124
1882 12.864 1922 12.787 1962 13.129
1883 12.783 1923 12.821 1963 13.16
1884 12.73 1924 12.764 1964 12.868
1885 12.754 1925 12.868 1965 12.935
1886 12.826 1926 13.014 1966 13.035
1887 12.723 1927 12.904 1967 13.031
1888 12.783 1928 12.871 1968 13.004
1889 12.922 1929 12.718 1969 13.117
1890 12.703 1930 12.964 1970 13.064
1891 12.767 1931 13.041 1971 12.903
1892 12.671 1932 12.992 1972 13.031
1893 12.631 1933 12.857 1973 13.175
1894 12.709 1934 12.982 1974 12.912
1895 12.728 1935 12.943 1975 12.975
1896 12.93 1936 12.993 1976 12.869
1897 12.936 1937 13.092 1977 13.148
1898 12.759 1938 13.187 1978 13.057
1899 12.874 1939 13.111 1979 13.154
1900 12.959 1940 13.055 1980 13.195

Data Set 5: Price of Petrol

The price of petrol in South Africa from August 1998 to July 2000 is shown in Table 5.

Table 5: Petrol prices
Date Price (R/l)
August 1998 R 2.37
September 1998 R 2.38
October 1998 R 2.35
November 1998 R 2.29
December 1998 R 2.31
January 1999 R 2.25
February 1999 R 2.22
March 1999 R 2.25
April 1999 R 2.31
May 1999 R 2.49
June 1999 R 2.61
July 1999 R 2.61
August 1999 R 2.62
September 1999 R 2.75
October 1999 R 2.81
November 1999 R 2.86
December 1999 R 2.85
January 2000 R 2.86
February 2000 R 2.81
March 2000 R 2.89
April 2000 R 3.03
May 2000 R 3.18
June 2000 R 3.22
July 2000 R 3.36

Grouping Data

One of the first steps to processing a large set of raw data is to arrange the data values together into a smaller number of groups, and then count how many of each data value there are in each group. The groups are usually based on some sort of interval of data values, so data values that fall into a specific interval, would be grouped together. The grouped data is often presented graphically or in a frequency table. (Frequency means “how many times”)

Exercise 1: Grouping Data

Group the elements of Data Set 1 to determine how many times the coin landed heads-up and how many times the coin landed tails-up.

Solution

  1. Step 1. Identify the groups :

    There are two unique data values: H and T. Therefore there are two groups, one for the H-data values and one for the T-data values.

  2. Step 2. Count how many data values fall into each group. :
    Table 6
    Data Value Frequency
    H 44
    T 56
  3. Step 3. Check that the total of the frequency column is equal to the total number of data values. :

    There are 100 data values and the total of the frequency column is 44+56=100.

Exercises - Grouping Data

  1. The height of 30 learners are given below. Fill in the grouped data below. (Tally is a convenient way to count in 5's. We use llll to indicate 5.)
    Table 7
    142163169132139140152168139150
    161132162172146152150132157133
    141170156155169138142160164168
    Table 8
    GroupTallyFrequency
    130 ≤h<≤h< 140  
    140 ≤h<≤h< 150  
    150 ≤h<≤h< 160  
    160 ≤h<≤h< 170  
    170 ≤h<≤h< 180  
    Click here for the solution
  2. An experiment was conducted in class and 50 learners were asked to guess the number of sweets in a jar. The following guesses were recorded.
    Table 9
    56494011333337293059
    21163844385222243034
    42154833514433171944
    47232747132553572823
    36354023453932582240
    Draw up a grouped frequency table using intervals 11-20, 21-30, 31-40, etc.
    Click here for the solution

Graphical Representation of Data

Once the data has been collected, it must be organised in a manner that allows for the information to be extracted most efficiently. One method of organisation is to display the data in the form of graphs. Functions and graphs have been studied in Functions and Graphs, and similar techniques will be used here. However, instead of drawing graphs from equations as was done in Functions and graphs, bar graphs, histograms and pie charts will be drawn directly from the data.

Bar and Compound Bar Graphs

A bar chart is used to present data where each observation falls into a specific category and where the categories, this is often for qualitative data. The frequencies (or percentages) are listed along the yy-axis and the categories are listed along the xx-axis. The heights of the bars correspond to the frequencies. The bars are of equal width and should not touch neighbouring bars.

A compound bar chart (also called component bar chart) is a variant: here the bars are cut into various components depending on what is being shown. If percentages are used for various components of a compound bar, then the total bar height must be 100%. The compound bar chart is a little more complex but if this method is used sensibly, a lot of information can be quickly shown in an attractive fashion.

Examples of a bar and a compound bar graph, for Data Set 1 , are shown in Figure 3. According to the frequency table for Data Set 1, the coin landed heads-up 44 times and tails-up 56 times.

Figure 3: Examples of a bar graph (left) and compound bar graph (right) for Data Set 1. The compound bar graph extends from 0% to 100%.
Figure 3 (MG10C16_003.png)

Histograms and Frequency Polygons

It is often useful to look at the frequency with which certain values fall in pre-set groups or classes of specified sizes. The choice of the groups should be such that they help highlight features in the data. If these grouped values are plotted in a manner similar to a bar graph, then the resulting graph is known as a histogram. Examples of histograms are shown in Figure 4 for Data Set 2, with group sizes of 1 and 2.

Table 10: Frequency table for Data Set 2, with a group size of 1.
Groups 0<n≤0<n≤1 1<n≤1<n≤2 2<n≤2<n≤3 3<n≤3<n≤4 4<n≤4<n≤5 5<n≤5<n≤6
Frequency 30 32 35 34 37 32
Table 11: Frequency table for Data Set 2, with a group size of 2.
Groups 0<n≤0<n≤2 2<n≤2<n≤4 4<n≤4<n≤6
Frequency 62 69 69
Figure 4: Examples of histograms for Data Set 2, with a group size = 1 (left) and a group size = 2 (right). The scales on the yy-axis for each graph are the same, and the values in the graph on the right are higher than the values of the graph on the left.
Figure 4 (MG10C16_004.png)

The same data used to plot a histogram are used to plot a frequency polygon, except the pair of data values are plotted as a point and the points are joined with straight lines. The frequency polygons for the histograms in Figure 4 are shown in Figure 5.

Figure 5: Examples of histograms for Data Set 2, with a group size = 1 (left) and a group size = 2 (right). The scales on the yy-axis for each graph are the same, and the values in the graph on the right are higher than the values of the graph on the left.
Figure 5 (MG10C16_005.png)

Unlike histograms, many frequency polygons can be plotted together to compare several frequency distributions, provided that the data has been grouped in the same way and provide a clear way to compare multiple datasets.

Pie Charts

A pie chart is a graph that is used to show what categories make up a specific section of the data, and what the contribution each category makes to the entire set of data. A pie chart is based on a circle, and each category is represented as a wedge of the circle or alternatively as a slice of the pie. The area of each wedge is proportional to the ratio of that specific category to the total number of data values in the data set. The wedges are usually shown in different colours to make the distinction between the different categories easier.

Figure 6: Example of a pie chart for Data Set 1. Pie charts show what contribution each group makes to the total data set.
Figure 6 (MG10C16_006.png)

Method: Drawing a pie-chart

  1. Draw a circle that represents the entire data set.
  2. Calculate what proportion of 360∘∘  each category corresponds to according to
    Angular Size = Frequency Total ×360∘ Angular Size = Frequency Total ×360∘
    (1)
  3. Draw a wedge corresponding to the angular contribution.
  4. Check that the total degrees for the different wedges adds up to close to 360∘360∘.

Exercise 2: Pie Chart

Draw a pie chart for Data Set 2, showing the relative proportions of each data value to the total.

Solution
  1. Step 1. Determine the frequency table for Data Set 2. :
    Table 12
      Total
    Data Value 1 2 3 4 5 6 –
    Frequency 30 32 35 34 37 32 200
  2. Step 2. Calculate the angular size of the wedge for each data value :
    Table 13
    Data Value Angular Size of Wedge
    1 Frequency Total × 360 ∘ = 30 200 × 360 = 54 ∘ Frequency Total × 360 ∘ = 30 200 × 360 = 54 ∘
    2 Frequency Total × 360 ∘ = 32 200 × 360 = 57 , 6 ∘ Frequency Total × 360 ∘ = 32 200 × 360 = 57 , 6 ∘
    3 Frequency Total × 360 ∘ = 35 200 × 360 = 63 ∘ Frequency Total × 360 ∘ = 35 200 × 360 = 63 ∘
    4 Frequency Total × 360 ∘ = 34 200 × 360 = 61 , 2 ∘ Frequency Total × 360 ∘ = 34 200 × 360 = 61 , 2 ∘
    5 Frequency Total × 360 ∘ = 37 200 × 360 = 66 , 6 ∘ Frequency Total × 360 ∘ = 37 200 × 360 = 66 , 6 ∘
    6 Frequency Total × 360 ∘ = 32 200 × 360 = 57 , 6 ∘ Frequency Total × 360 ∘ = 32 200 × 360 = 57 , 6 ∘
  3. Step 3. Draw the pie, with the size of each wedge as calculated above. :

    Figure 7
    Figure 7 (MG10C16_007.png)

Note that the total angular size of the wedges may not add up to exactly 360∘∘  because of rounding.

Line and Broken Line Graphs

All graphs that have been studied until this point (bar, compound bar, histogram, frequency polygon and pie) are drawn from grouped data. The graphs that will be studied in this section are drawn from the ungrouped or raw data.

Line and broken line graphs are plots of a dependent variable as a function of an independent variable, e.g. the average global temperature as a function of time, or the average rainfall in a country as a function of season.

Usually a line graph is plotted after a table has been provided showing the relationship between the two variables in the form of pairs. Just as in (x,y) graphs, each of the pairs results in a specific point on the graph, and being a line graph these points are connected to one another by a line.

Many other line graphs exist; they all connect the points by lines, not necessarily straight lines. Sometimes polynomials, for example, are used to describe approximately the basic relationship between the given pairs of variables, and between these points.

Figure 8: Example of a line graph for Data Set 5.
Figure 8 (MG10C16_008.png)

Exercise 3: Line Graphs

Clawde the cat is overweight and her owners have decided to put her on a restricted eating plan. Her mass is measured once a month and is tabulated below. Draw a line graph of the data to determine whether the restricted eating plan is working.

Table 14
Month Mass (kg)
March 4,53
April 4,56
May 4,51
June 4,41
July 4,41
August 4,36
September 4,43
October 4,37
Solution
  1. Step 1. Determine what is required :

    We are required to plot a line graph to determine whether the restricted eating plan is helping Clawde the cat lose weight. We are given all the information that we need to plot the graph.

  2. Step 2. Plot the graph :

    Figure 9
    Figure 9 (MG10C16_009.png)

  3. Step 3. Analyse Graph :

    There is a slight decrease of mass from March to October, so the restricted eating plan is working, but very slowly.

Exercises - Graphical Representation of Data

  1. Represent the following information on a pie chart.
    Table 15
    Walk15
    Cycle24
    Train18
    Bus8
    Car35
    Total100
    Click here for the solution
  2. Represent the following information using a broken line graph.
    Table 16
    Time07h0008h0009h0010h0011h0012h00
    Temp (∘∘C)1616,517192024
    Click here for the solution
  3. Represent the following information on a histogram. Using a coloured pen, draw a frequency polygon on this histogram.
    Table 17
    Time in secondsFrequency
    16 - 255
    26 - 3510
    36 - 4526
    46 - 5530
    56 - 6515
    66 - 7512
    76 - 8510
    Click here for the solution
  4. The maths marks of a class of 30 learners are given below, represent this information using a suitable graph.
    Table 18
    82756654797829556891
    43489061456082637253
    51326242496281496160
    Click here for the solution
  5. Use a compound bar graph to illustrate the following information
    Table 19
    Year20032004200520062007
    Girls1815131215
    Boys1511181610
    Click here for the solution

Summarising Data

If the data set is very large, it is useful to be able to summarise the data set by calculating a few quantities that give information about how the data values are spread and about the central values in the data set.

Measures of Central Tendency

Mean or Average

The mean, (also known as arithmetic mean), is simply the arithmetic average of a group of numbers (or data set) and is shown using the bar symbol ¯¯. So the mean of the variable xx is x¯x¯ pronounced "x-bar". The mean of a set of values is calculated by adding up all the values in the set and dividing by the number of items in that set. The mean is calculated from the raw, ungrouped data.

Definition 2: Mean

The mean of a data set, xx, denoted by x¯x¯, is the average of the data values, and is calculated as:

x ¯ = sum of all values number of values = x 1 + x 2 + x 3 + ... + x n n x ¯ = sum of all values number of values = x 1 + x 2 + x 3 + ... + x n n
(2)

Method: Calculating the mean

  1. Find the total of the data values in the data set.
  2. Count how many data values there are in the data set.
  3. Divide the total by the number of data values.
Exercise 4: Mean

What is the mean of x={10,20,30,40,50}x={10,20,30,40,50}?

Solution
  1. Step 1. Find the total of the data values :
    10 + 20 + 30 + 40 + 50 = 150 10 + 20 + 30 + 40 + 50 = 150
    (3)
  2. Step 2. Count the number of data values in the data set :

    There are 5 values in the data set.

  3. Step 3. Divide the total by the number of data values. :
    150 ÷ 5 = 30 150 ÷ 5 = 30
    (4)
  4. Step 4. Answer :

    ∴∴ the mean of the data set x={10,20,30,40,50}x={10,20,30,40,50} is 30.

Median

Definition 3: Median

The median of a set of data is the data value in the central position, when the data set has been arranged from highest to lowest or from lowest to highest. There are an equal number of data values on either side of the median value.

The median is calculated from the raw, ungrouped data, as follows.

Method: Calculating the median

  1. Order the data from smallest to largest or from largest to smallest.
  2. Count how many data values there are in the data set.
  3. Find the data value in the central position of the set.
Exercise 5: Median

What is the median of {10,14,86,2,68,99,1}{10,14,86,2,68,99,1}?

Solution
  1. Step 1. Order the data set from lowest to highest :

    1,2,10,14,68,86,99

  2. Step 2. Count the number of data values in the data set :

    There are 7 points in the data set.

  3. Step 3. Find the central position of the data set :

    The central position of the data set is 4.

  4. Step 4. Find the data value in the central position of the ordered data set. :

    14 is in the central position of the data set.

  5. Step 5. Answer :

    ∴∴ 14 is the median of the data set {1,2,10,14,68,86,99}{1,2,10,14,68,86,99}.

This example has highlighted a potential problem with determining the median. It is very easy to determine the median of a data set with an odd number of data values, but what happens when there is an even number of data values in the data set?

When there is an even number of data values, the median is the mean of the two middle points.

Tip:
Finding the Central Position of a Data Set

An easy way to determine the central position or positions for any ordered data set is to take the total number of data values, add 1, and then divide by 2. If the number you get is a whole number, then that is the central position. If the number you get is a fraction, take the two whole numbers on either side of the fraction, as the positions of the data values that must be averaged to obtain the median.

Exercise 6: Median

What is the median of {11,10,14,86,2,68,99,1}{11,10,14,86,2,68,99,1}?

Solution
  1. Step 1. Order the data set from lowest to highest :

    1,2,10,11,14,68,85,99

  2. Step 2. Count the number of data values in the data set :

    There are 8 points in the data set.

  3. Step 3. Find the central position of the data set :

    The central position of the data set is between positions 4 and 5.

  4. Step 4. Find the data values around the central position of the ordered data set. :

    11 is in position 4 and 14 is in position 5.

  5. Step 5. Answer :

    ∴∴ the median of the data set {1,2,10,11,14,68,85,99}{1,2,10,11,14,68,85,99} is

    ( 11 + 14 ) ÷ 2 = 12 , 5 ( 11 + 14 ) ÷ 2 = 12 , 5
    (5)

Mode

Definition 4: Mode

The mode is the data value that occurs most often, i.e. it is the most frequent value or most common value in a set.

Method: Calculating the mode Count how many times each data value occurs. The mode is the data value that occurs the most.

The mode is calculated from grouped data, or single data items.

Exercise 7: Mode

Find the mode of the data set x={1,2,3,4,4,4,5,6,7,8,8,9,10,10}x={1,2,3,4,4,4,5,6,7,8,8,9,10,10}

Solution
  1. Step 1. Count how many times each data value occurs. :
    Table 20
    data value frequency data value frequency
    1 1 6 1
    2 1 7 1
    3 1 8 2
    4 3 9 1
    5 1 10 2
  2. Step 2. Find the data value that occurs most often. :

    4 occurs most often.

  3. Step 3. Answer :

    The mode of the data set x={1,2,3,4,4,4,5,6,7,8,8,9,10,10}x={1,2,3,4,4,4,5,6,7,8,8,9,10,10} is 4. Since the number 4 appears the most frequently.

A data set can have more than one mode. For example, both 2 and 3 are modes in the set 1, 2, 2, 3, 3. If all points in a data set occur with equal frequency, it is equally accurate to describe the data set as having many modes or no mode.

Figure 10
Khan academy video on statistics

Measures of Dispersion

The mean, median and mode are measures of central tendency, i.e. they provide information on the central data values in a set. When describing data it is sometimes useful (and in some cases necessary) to determine the spread of a distribution. Measures of dispersion provide information on how the data values in a set are distributed around the mean value. Some measures of dispersion are range, percentiles and quartiles.

Range

Definition 5: Range

The range of a data set is the difference between the lowest value and the highest value in the set.

Method: Calculating the range

  1. Find the highest value in the data set.
  2. Find the lowest value in the data set.
  3. Subtract the lowest value from the highest value. The difference is the range.
Exercise 8: Range

Find the range of the data set x={1,2,3,4,4,4,5,6,7,8,8,9,10,10}x={1,2,3,4,4,4,5,6,7,8,8,9,10,10}

Solution
  1. Step 1. Find the highest and lowest values. :

    10 is the highest value and 1 is the lowest value.

  2. Step 2. Subtract the lowest value from the highest value to calculate the range. :
    10 - 1 = 9 10 - 1 = 9
    (6)
  3. Step 3. Answer :

    For the data set x={1,2,3,4,4,4,5,6,7,8,8,9,10,10}x={1,2,3,4,4,4,5,6,7,8,8,9,10,10}, the range is 9.

Quartiles

Definition 6: Quartiles

Quartiles are the three data values that divide an ordered data set into four groups containing equal numbers of data values. The median is the second quartile.

The quartiles of a data set are formed by the two boundaries on either side of the median, which divide the set into four equal sections. The lowest 25% of the data being found below the first quartile value, also called the lower quartile. The median, or second quartile divides the set into two equal sections. The lowest 75% of the data set should be found below the third quartile, also called the upper quartile. For example:

Table 21
22 24 48 51 60 72 73 75 80 88 90
    ↓ ↓     ↓ ↓     ↓ ↓    
    Lower quartile     Median     Upper quartile    
    (Q1Q1)     (Q2Q2)     (Q3Q3)    

Method: Calculating the quartiles

  1. Order the data from smallest to largest or from largest to smallest.
  2. Count how many data values there are in the data set.
  3. Divide the number of data values by 4. The result is the number of data values per group.
  4. Determine the data values corresponding to the first, second and third quartiles using the number of data values per quartile.
Exercise 9: Quartiles

What are the quartiles of {3,5,1,8,9,12,25,28,24,30,41,50}{3,5,1,8,9,12,25,28,24,30,41,50}?

Solution
  1. Step 1. Order the data set from lowest to highest :

    { 1 , 3 , 5 , 8 , 9 , 12 , 24 , 25 , 28 , 30 , 41 , 50 } { 1 , 3 , 5 , 8 , 9 , 12 , 24 , 25 , 28 , 30 , 41 , 50 }

  2. Step 2. Count the number of data values in the data set :

    There are 12 values in the data set.

  3. Step 3. Divide the number of data values by 4 to find the number of data values per quartile. :
    12 ÷ 4 = 3 12 ÷ 4 = 3
    (7)
  4. Step 4. Find the data values corresponding to the quartiles. :
    Table 22
    1 3 5 ∥ ∥ 8 9 12 ∥ ∥ 24 25 28 ∥ ∥ 30 41 50
          Q 1 Q 1       Q 2 Q 2       Q 3 Q 3      

    The first quartile occurs between data position 3 and 4 and is the average of data values 5 and 8. The second quartile occurs between positions 6 and 7 and is the average of data values 12 and 24. The third quartile occurs between positions 9 and 10 and is the average of data values 28 and 30.

  5. Step 5. Answer :

    The first quartile = 6,5. (Q1Q1)

    The second quartile = 18. (Q2Q2)

    The third quartile = 29. (Q3Q3)

Inter-quartile Range

Definition 7: Inter-quartile Range

The inter quartile range is a measure which provides information about the spread of a data set, and is calculated by subtracting the first quartile from the third quartile, giving the range of the middle half of the data set, trimming off the lowest and highest quarters, i.e. Q3-Q1Q3-Q1.

The semi-interquartile range is half the interquartile range, i.e. Q3-Q12Q3-Q12

Exercise 10: Medians, Quartiles and the Interquartile Range

A class of 12 students writes a test and the results are as follows: 20, 39, 40, 43, 43, 46, 53, 58, 63, 70, 75, 91. Find the range, quartiles and the Interquartile Range.

Solution
  1. Step 1. :
    Table 23
    20 39 40 ∥ ∥ 43 43 46 ∥ ∥ 53 58 63 ∥ ∥ 70 75 91
          Q 1 Q 1       M M       Q 3 Q 3      
  2. Step 2. The Range :

    The range = 91 - 20 = 71. This tells us that the marks are quite widely spread.

  3. Step 3. The median lies between the 6th and 7th mark :

    i.e. M=46+532=992=49,5M=46+532=992=49,5

  4. Step 4. The lower quartile lies between the 3rd and 4th mark :

    i.e. Q1=40+432=832=41,5Q1=40+432=832=41,5

  5. Step 5. The upper quartile lies between the 9th and 10th mark :

    i.e. Q3=63+702=1332=66,5Q3=63+702=1332=66,5

  6. Step 6. Analysing the quartiles :

    The quartiles are 41,5, 49,5 and 66,5. These quartiles tell us that 25%% of the marks are less than 41,5; 50%% of the marks are less than 49,5 and 75%% of the marks are less than 66,5. They also tell us that 50%% of the marks lie between 41,5 and 66,5.

  7. Step 7. The Interquartile Range :

    The Interquartile Range = 66,5 - 41,5 = 25. This tells us that the width of the middle 50%% of the data values is 25.

  8. Step 8. The Semi-interquatile Range :

    The Semi-interquartile Range = 252252 = 12,5

Percentiles

Definition 8: Percentiles

Percentiles are the 99 data values that divide a data set into 100 groups.

The calculation of percentiles is identical to the calculation of quartiles, except the aim is to divide the data values into 100 groups instead of the 4 groups required by quartiles.

Method: Calculating the percentiles

  1. Order the data from smallest to largest or from largest to smallest.
  2. Count how many data values there are in the data set.
  3. Divide the number of data values by 100. The result is the number of data values per group.
  4. Determine the data values corresponding to the first, second and third quartiles using the number of data values per quartile.

Exercises - Summarising Data

  1. Three sets of data are given:
    1. Data set 1: 9 12 12 14 16 22 24
    2. Data set 2: 7 7 8 11 13 15 16 16
    3. Data set 3: 11 15 16 17 19 19 22 24 27 For each one find:
      1. the range
      2. the lower quartile
      3. the interquartile range
      4. the semi-interquartile range
      5. the median
      6. the upper quartile
    Click here for the solution
  2. There is 1 sweet in one jar, and 3 in the second jar. The mean number of sweets in the first two jars is 2.
    1. If the mean number in the first three jars is 3, how many are there in the third jar?
    2. If the mean number in the first four jars is 4, how many are there in the fourth jar?
    Click here for the solution
  3. Find a set of five ages for which the mean age is 5, the modal age is 2 and the median age is 3 years.
    Click here for the solution
  4. Four friends each have some marbles. They work out that the mean number of marbles they have is 10. One of them leaves. She has 4 marbles. How many marbles do the remaining friends have together?
    Click here for the solution

Exercise 11: Mean, Median and Mode for Grouped Data

Consider the following grouped data and calculate the mean, the modal group and the median group.

Table 24
Mass (kg) Frequency
41 - 45 7
46 - 50 10
51 - 55 15
56 - 60 12
61 - 65 6
  Total = 50
Solution
  1. Step 1. Calculating the mean :

    To calculate the mean we need to add up all the masses and divide by 50. We do not know actual masses, so we approximate by choosing the midpoint of each group. We then multiply those midpoint numbers by the frequency. Then we add these numbers together to find the approximate total of the masses. This is show in the table below.

    Table 25
    Mass (kg) Midpoint Frequency Midpt ×× Freq
    41 - 45 (41+45)/2 = 43 7 43 ×× 7 = 301
    46 - 50 48 10 480
    51 - 55 53 15 795
    56 - 60 58 12 696
    61 - 65 63 6 378
        Total = 50 Total = 2650
  2. Step 2. Answer :

    The mean = 265050=53265050=53.

    The modal group is the group 51 - 53 because it has the highest frequency.

    The median group is the group 51 - 53, since the 25th and 26th terms are contained within this group.

More mean, modal and median group exercises.

In each data set given, find the mean, the modal group and the median group.

  1. Times recorded when learners played a game.
    Table 26
    Time in secondsFrequency
      
    36 - 455
    46 - 5511
    56 - 6515
    66 - 7526
    76 - 8519
    86 - 9513
    96 - 1056
    Click here for the solution
  2. The following data were collected from a group of learners.
    Table 27
    Mass in kilogramsFrequency
      
    41 - 453
    46 - 505
    51 - 558
    56 - 6012
    61 - 6514
    66 - 709
    71 - 757
    76 - 802
    Click here for the solution

Misuse of Statistics

In many cases groups can gain an advantage by misleading people with the misuse of statistics.

Common techniques used include:

  • Three dimensional graphs.
  • Axes that do not start at zero.
  • Axes without scales.
  • Graphic images that convey a negative or positive mood.
  • Assumption that a correlation shows a necessary causality.
  • Using statistics that are not truly representative of the entire population.
  • Using misconceptions of mathematical concepts

For example, the following pairs of graphs show identical information but look very different. Explain why.

Figure 11
Figure 11 (MG10C16_010.png)

Exercises - Misuse of Statistics

  1. A company has tried to give a visual representation of the increase in their earnings from one year to the next. Does the graph below convince you? Critically analyse the graph.
    Figure 12
    Figure 12 (MG10C16_011.png)
    Click here for the solution
  2. In a study conducted on a busy highway, data was collected about drivers breaking the speed limit and the colour of the car they were driving. The data were collected during a 20 minute time interval during the middle of the day, and are presented in a table and pie chart below.
    • Conclusions made by a novice based on the data are summarised as follows:
    • “People driving white cars are more likely to break the speed limit.”
    • “Drivers in blue and red cars are more likely to stick to the speed limit.”
    • Do you agree with these conclusions? Explain.
    Click here for the solution
  3. A record label produces a graphic, showing their advantage in sales over their competitors. Identify at least three devices they have used to influence and mislead the readers impression.
    Figure 13
    Figure 13 (MG10C16_013.png)
    Click here for the solution
  4. In an effort to discredit their competition, a tour bus company prints the graph shown below. Their claim is that the competitor is losing business. Can you think of a better explanation?
    Figure 14
    Figure 14 (MG10C16_014.png)
    Click here for the solution
  5. To test a theory, 8 different offices were monitored for noise levels and productivity of the employees in the office. The results are graphed below.
    Figure 15
    Figure 15 (MG10C16_015.png)
    The following statement was then made: “If an office environment is noisy, this leads to poor productivity.” Explain the flaws in this thinking.
    Click here for the solution

Summary of Definitions

  • mean: The mean of a data set, xx, denoted by x¯x¯, is the average of the data values, and is calculated as:
    x¯=sum of values number of values x¯=sum of values number of values
    (8)
  • median: The median is the centre data value in a data set that has been ordered from lowest to highest
  • mode: The mode is the data value that occurs most often in a data set.

The following presentation summarises what you have learnt in this chapter. Ignore the chapter number and any exercise numbers in the presentation.

Figure 16

Exercises

  1. Calculate the mean, median, and mode of Data Set 3.
    Click here for the solution
  2. The tallest 7 trees in a park have heights in metres of 41, 60, 47, 42, 44, 42, and 47. Find the median of their heights.
    Click here for the solution
  3. The students in Bjorn's class have the following ages: 5, 9, 1, 3, 4, 6, 6, 6, 7, 3. Find the mode of their ages.
    Click here for the solution
  4. The masses (in kg, correct to the nearest 0,1 kg) of thirty people were measured as follows:
    Table 28
    45,157,967,957,450,761,163,967,569,771,7
    68,063,258,756,978,559,754,466,451,647,7
    70,954,859,160,360,152,674,972,149,549,8
    1. Copy the frequency table below, and complete it.
      Table 29
      Mass (in kg)TallyNumber of people
      45,0≤m<50,045,0≤m<50,0  
      50,0≤m<55,050,0≤m<55,0  
      55,0≤m<60,055,0≤m<60,0  
      60,0≤m<65,060,0≤m<65,0  
      65,0≤m<70,065,0≤m<70,0  
      70,0≤m<75,070,0≤m<75,0  
      75,0≤m<80,075,0≤m<80,0  
    2. Draw a frequency polygon for this information.
    3. What can you conclude from looking at the graph?
    Click here for the solution
  5. An engineering company has designed two different types of engines for motorbikes. The two different motorbikes are tested for the time it takes (in seconds) for them to accelerate from 0 km/h to 60 km/h.
    Table 30
     Test 1Test 2Test 3Test 4Test 5Test 6Test 7Test 8Test 9Test 10Average
    Bike 11.551.000.920.801.490.711.060.680.871.09 
    Bike 20.91.01.11.01.00.90.91.00.91.1 
    1. What measure of central tendency should be used for this information?
    2. Calculate the average you chose in the previous question for each motorbike.
    3. Which motorbike would you choose based on this information? Take note of accuracy of the numbers from each set of tests.
    Click here for the solution
  6. The heights of 40 learners are given below.
    Table 31
    154140145159150132149150138152
    141132169173139161163156157171
    168166151152132142170162146152
    142150161138170131145146147160
    1. Set up a frequency table using 6 intervals.
    2. Calculate the approximate mean.
    3. Determine the mode.
    4. How many learners are taller than your approximate average in (b)?
    Click here for the solution
  7. In a traffic survey, a random sample of 50 motorists were asked the distance they drove to work daily. This information is shown in the table below.
    Table 32
    Distance in km1-56-1011-1516-2021-2526-3031-3536-4041-45
    Frequency4591078322
    1. Find the approximate mean.
    2. What percentage of samples drove
      1. less than 16 km?
      2. more than 30 km?
      3. between 16 km and 30 km daily?
    Click here for the solution
  8. A company wanted to evaluate the training programme in its factory. They gave the same task to trained and untrained employees and timed each one in seconds.
    Table 33
    Trained121137131135130
     128130126132127
     129120118125134
    Untrained135142126148145
     156152153149145
     144134139140142
    1. Find the medians and quartiles for both sets of data.
    2. Find the Interquartile Range for both sets of data.
    3. Comment on the results.
    Click here for the solution
  9. A small firm employs nine people. The annual salaries of the employers are:
    Table 34
    R600 000R250 000R200 000
    R120 000R100 000R100 000
    R100 000R90 000R80 000
    1. Find the mean of these salaries.
    2. Find the mode.
    3. Find the median.
    4. Of these three figures, which would you use for negotiating salary increases if you were a trade union official? Why?
    Click here for the solution
  10. The marks for a particular class test are listed here:
    Table 35
    67589167588271516084
    31679664787187788938
    6962607360877149  

    Complete the frequency table using the given class intervals.

    Table 36
    ClassTallyFrequencyMid-pointFreq ×× Midpt
    30-39 34,5  
    40-49 44,5  
    50-59    
    60-69    
    70-79    
    80-89    
    90-99    
      Sum = Sum =

    Click here for the solution

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