1331.0 - Statistics - A Powerful Edge!, 1996  
ARCHIVED ISSUE Released at 11:30 AM (CANBERRA TIME) 31/07/1998   
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Contents >> Stats Maths >> Measures of Location - Mean

MEASURES OF LOCATION

The centre of a set of data is important. Often, you want to know what most people think, or the average of a set of values. If you have a normal distribution (see section Stem and Leaf Plots), a measure of the centre provides information on what the value is for most of the population.

Finding the central location of a data set requires the calculation of the mean, median or mode. All three measures indicate the location of the centre (often called the central tendency). However, each measure has its own definition and application in different situations.


MEAN

DEFINITION

The mean of a numeric variable is calculated by summing the values of all observations in a data set and then dividing by the number of observations in the set. It is often referred to as the average. Thus:

MEAN = SUM OF ALL THE OBSERVATION VALUES ÷ NUMBER OF OBSERVATIONS

DISCRETE VARIABLES

EXAMPLE
1.In 1997 Tony Modra was a leading goal kicker in the Australian Football League. In 10 matches he kicked 7, 5, 0, 7, 8, 5, 5, 4, 5, & 1 goals. What was his mean score?

Mean = sum of all the observed values ÷ number of observations

= (7+5+0+7+8+5+5+4+5+1) ÷ 10

= 47 ÷ 10

= 4.7

Therefore, for the above 10 matches Tony Modra kicked an average 4.7 goals per match. The value 4.7 is not a whole number so it only has meaning in a statistical sense. In reality it is impossible to kick 4.7 goals (even if you are Tony Modra). Note: it is possible to kick 6.17 goals. Why?

Using mathematical notation, for a discrete variable the mean is calculated as follows:

Equation: the mean of a discrete varible.
Image: what the letters stand for in the equation of the mean of a discrete varible.

2.The number of people killed in road traffic accidents in New South Wales from 1983 to 1996 is given in the following table. What was the average number of people killed per year on New South Wales’ roads from 1983 to 1996? How many people died daily in road traffic accidents in New South Wales during this period?

Year
People killed
1987
959
1988
1,037
1989
960
1990
797
1991
663
1993
652
1994
560
1995
623
1996
583

Equation: the mean of a discrete varible.

= 7,453 ÷ 10

= 745.3

This is the average number of people killed per year on New South Wales’ roads from 1987 to 1996.

To calculate the daily death rate from road traffic accidents, the average yearly death rate is divided by the number of days in a year (leap years are ignored).

Thus: 745.3 ÷ 365 = 2.0 deaths/day approximately

Therefore, on average, approximately 2.0 people died daily in road traffic accidents in New South Wales from 1987 to 1996.

Historical note: the highest road toll recorded in New South Wales was in 1978 when 1,384 people lost their lives.

How do you think road traffic accident statistics can be used to reduce the number of people killed on the roads each year?


FREQUENCY TABLE (DISCRETE VARIABLES)
3.Grouping observations in tables is useful when dealing with a large amount of data. Tony Modra’s goal kicking figures can be displayed in a frequency table:

No. of goals (x)
Frequency(f)
xf
0
1
0
1
1
1
4
1
4
5
4
20
7
2
14
8
1
8

Because the observations are grouped, the mathematical notation changes slightly. For a discrete variable in a frequency table the mean is calculated as follows:
Equation: the mean of a discrete varible in a frequency table
Image: what the letters mean in the equation of the mean of a discrete varible in a frequency table

Image: the mean of a discrete varible in a frequency table worked out
    GROUPED VARIABLES (CONTINUOUS OR DISCRETE)

    NOTE:

    Determine the mid-point of each class interval for a variable before calculating the mean from a frequency table. This method provides an approximation of the true mean for an ungrouped variable. How good the approximation is depends on how evenly the observed values are spread within each group.


    4.The following table shows the heights of 50 randomly selected Year 10 girls in a school. What is the mean height of the girls?

    Height (centimetres)
    Mid-point (x)
    Frequency (f)
    xf
    150 - <155
    152.5
    4
    610.0
    155 - <160
    157.5
    7
    1,102.5
    160 - <165
    162.5
    18
    2,925.0
    165 - <170
    167.5
    11
    1,842.5
    170 - <175
    172.5
    6
    1,035.0
    175 - <180
    177.5
    4
    710.0
    -
    -
    50
    8,225.0
      Thus:
      Equation: the mean of a discrete varible in a frequency table

      = 8,225 /50

      =      164.5 cm

    Therefore, the mean height of the 50 Year 10 girls is 164.5 cm.



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