Central Tendency

DontbeAverage.jpgWhen a sample has been taken from a population, the data has to be analysed.

Different symbols are used for samples and populations.

A parameter is a quantity measured from a population.
A statistic is a quantity measured from a sample.

 
Population Parameter
Sample Statistic
Mean
μ
Y12_Central_Tendency_01.gif
Standard deviation
σ
s

Types of Data

Discrete data This is data that is usually whole numbers and is often collected by counting

e.g. The number of people at a sports game or the number of cars sold.

Continuous data This data is usually the result of measuring and can be any type of number.

e.g. The heights of people or the weights of passengers' bags at an airport.

An average is a number that represents the centre or central tendency of a set of data, or is typical of the sample. 
Three types of average are commonly used: The mean, the median and the mode.

The Mean of a Sample

The mean is commonly known by most people as the average.

The symbol used for the mean of a sample is .

The mean is calculated by adding together all of the scores or values and dividing by the number of scores or values.

Y12_Central_Tendency_02.gif

If the data is in a frequency table, each score is multiplied by its frequency.

Y12_Central_Tendency_03.gif

Finding the mean of a frequency distribution, including grouped data, is covered Topic 53, Statistical Graphs.

Findng the mean on a calculator

The procedure for finding statistical values will vary slightly from calculator to calculator.

For a typical scientific calculator, to find the mean of 5, 7, and 12:

Task
Press
Action
Select the statistical mode
MODE Y12_Central_Tendency_04.gif

Gives three choices: COMP(1), SD(2) or REG(3)

Selects SD (statistics) mode.
Enter the data
Y12_Central_Tendency_05.gif DT Y12_Central_Tendency_06.gif DT Y12_Central_Tendency_07.gif Y12_Central_Tendency_04.gifDT
 Enters the three numbers 5, 7 and 12.
Find mean
Y12_Central_Tendency_08.gif  Y12_Central_Tendency_07.gif  Y12_Central_Tendency_09.gif   8
 Finds the mean,  = 8
Clear old data
Y12_Central_Tendency_08.gif   Y12_Central_Tendency_10.gif  Y12_Central_Tendency_09.gif
 Always do this before entering new data.

 

Finding the mean on a spreadsheet

Enter the data:

Y12_Central_Tendency_11.gif

The function entered in cell B5 to find the mean is =AVERAGE(A2..A4) this gives a mean of 8.

The Median

The median is the middle score or value when the data is arranged in order.

If there are an even number of scores, the median is halfway between the middle two values.

Finding the median from a set of data is covered further in the Spread topic.

The Mode

The mode is the most common score or value.

If all of the scores are different, there is no mode.

If the data is grouped together in a frequency table, the group with highest frequency is called the modal group or class.

 

Examples
Answer

For the following set of test results of 10 people:

3, 8, 5, 6, 8, 5, 4, 1, 10, 5

Find:

(a) The mean

(b) The mode

(c) The median

 

(a) The mean

Y12_Central_Tendency_12.gif

(b) The mode = 5 (occurs 3 times)

(c) Arrange in order:

1, 3, 4, 5, 5, 5, 6, 8, 8, 10

Median is 5

 

Finding Averages in a Frequency Distribution

The table shows the lengths of 100 possums caught in traps.

Lengths of possums (cm)
Frequency(f)
0 -
7
10 -
18
20 -
20
30 -
33
40 -
12
50 − 60
10

Mode The modal interval is the most common length which is 30 cm − 40 cm.

Median The median is midway between the 50th − 51st lengths which would be in the interval 30 cm − 40 cm.

Mean To find the approximate mean from a frequency table of grouped data the midpoint of each interval is multiplied by its frequency.

Length of possum (cm)
Midpoint (x)
Frequency(f)
f .x
0 -
5
7
5 x 7 = 35
10 -
15
18
15 x 18 = 270
20 -
25
20
25 x 20 = 500
30 -
35
33
35 x 33 = 1155
40 -
45
12
45 x 12 = 540
50 − 60
55
10
55 × 10 = 550
Totals
  
100
3050

 

The approximate mean length of the 100 possums is 3050 ÷ 100 = 30.5 cm

Which average to use?

Average
Advantage
Disadvantage
mean

Uses all of the values.

Can be found easily on a calculator.

Influenced by extreme high or low values.

e.g. 5, 5, 6, 7, 8, 9, 100
Mean is 140÷7 =20 

The value of 100 has a big effect on the mean.
median

Not influenced by extreme high or low values.

e.g. 4, 5, 6, 7, 8, 9, 100

The value of 100 does not effect the median.

 Hard to work out if there are a large number of values.
mode

Good for finding the most popular value.

e.g. Finding the most popular pizza size.

May not be all that representative of a set of values.

e.g. 3, 3, 4, 5, 6, 8, 9, 10, 12.

The mode is 3 which is not near the middle.

The Working with Data activity provides practice at finding the mean, median and mode.