## Central Tendency When a sample has been taken from a population, the data has to be analysed.

Different symbols are used for samples and populations.

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

 Population Parameter Sample Statistic Mean μ 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. If the data is in a frequency table, each score is multiplied by its frequency. 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 Gives three choices: COMP(1), SD(2) or REG(3) Selects SD (statistics) mode. Enter the data DT DT  DT Enters the three numbers 5, 7 and 12. Find mean   8 Finds the mean, = 8 Clear old data   Always do this before entering new data.

Finding the mean on a spreadsheet

Enter the data: 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 (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, 100Mean 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.