AVERAGES

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Data can be reduced from a set of data to just one statistic, the centre, or measure of central location is known as the average.

Mean: or arithmetric mean is the most used average. To calculate, all the values in a set of data are summed and then divided by the number of separate values.

Example: Find the mean of: 6, 7, 4, 9, 7, 3.

Answer: Total = 36. Number of separate values = 6. Therefore 366. Mean = 6.

= sum of. x = values. n = number of values. Formula = .

Exercise: Find the mean.

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Mode: This is the most frequently occurring number in the set of data.

Example: Find the mode of: 6, 7, 4, 9, 7, 3.

Answer: 7 is the most frequently occurring number. If there are two numbers which appear the most frequent this is called bimodal.

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Median: This is the value of the middle number when the set is arranged in numerical order.

Example: Find the median of: 8, 3, 5, 25, 12, 15, 2, 9, 11.

Answer: rearrange: 2, 3, 5, 8, 9, 11, 12, 15, 25. The middle number is 9.

Example: Find the median of: 31, 44, 23, 16.

Answer: rearrange: 16, 23, 31,44. There are 2 middle numbers, take the average, (23 + 31 = 54. 54 2 = 27) therefore answer is 27.

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Range: The range of a set of numbers is the difference between the highest and lowest numbers.

Example: Find the range of the following numbers: 8, 5, 25, 12, 15, 11.

Answer: Highest: 25. Lowest: 5. Range: 20.

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Mean from Frequency Tables: When data is presented in the form of a table it saves a lot of time not having to repeat numbers.

Example: Find the mean from the frequency table.

Value:            1    2    3    4    5

Frequency:    6    5    4    3    2

Table:                 Value:  x          Frequency:  f        Value  x  Frequency:   xf

                   1                           6                           6
                                      2                           5                          10
                                      3                           4                          12
                                      4                           3                          12
                                      5                           2                          10

The totals: f = 20 (Total Frequency). xf = 50 (Total of 20 values).

The mean value: (xff) = 5020 = 2.

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Cumulative Frequency: At a value is the running total of all the frequencies up to that value. A cumulative frequency table gives the cumulative frequencies at each value.

Example: A survey of used cars for sale revealed the following prices:

Price Range:       under 500,     500-1000    1000-1500    1500-2000    2000-2500

Frequency:                  4                   12                  13                 10                   6

With this information a cumulative frequency table is constructed:

Prices:                                 Cumulative Frequency:

0 - 500                                        4
500 - 1000                                 12 + 4 = 16
1000 - 1500                               13 + 16 = 29
1500 - 2000                               10 + 29 = 39
2000 - 2500                                 6 + 39 = 45

A frequency curve is constructed by plotting the cumulative frequencies against the upper boundaries of the prices.

The median is found by taking a line from the centre of the frequencies and reading that value, here 1,220.

The interquartile range is the range between andof the cumulative frequency range, two lines taken and the values read, here it is between 840 and 1,700. The range being 1,700 - 840 = 860.

To see the Graph click : (28Kb).

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