CHAPTER 36

Averages and Range

Range and Averages RANGE

RANGE = LARGEST VALUE – SMALLEST VALUE

TYPES OF AVERAGE

1. The MOST COMMON value is the MODE.

2. When the values are arranged in order of size, from smallest to largest, the MIDDLE value is the MEDIAN. If there are an even number of values the MEDIAN is the MEAN of the TWO MIDDLE values.

3. The MEAN is calculated by ADDING all the values and

DIVIDING this TOTAL by the number of values.

Frequency Distributions To find the averages for a FREQUENCY DISTRIBUTION

we have to take the FREQUENCY associated with each value into account.

1. The MODE is the value with the LARGEST FREQUENCY

2. The MEDIAN is the MIDDLE value when the value are in ascending order, however it is important to consider the frequency associated with each value to ensure we find the MEDIAN correctly. In general the MEDIAN is at ½(∑f +1), where ∑f is the total frequency

3. MEAN = ∑fx∑f.

where ∑fx is the total of all the data values multiplied by their associated frequencyand ∑f is the total frequency

Averages for Frequency DistributionsEg Erin kept a record of the goals that the Slemish College

hockey team had scored. Work out the MEAN number of goals scored.

Goals(x)

Frequency(f)

Frequency x Goals(fx)

012345

456302

4 x 0 = 05 x 1 = 5

6 x 2 = 123 x 3 = 90 x 4 = 0

2 x 5 = 10

∑f = 20 ∑fx = 36

Averages for Frequency Distributions

The easy way to do this is to add another column to the table to work out the total number of goals scored. MEAN = ∑fx ∑f = 36 20 = 1.8 goalsWhere:∑fx is the sum of the goals∑f is the sum of the matches

Averages for Frequency DistributionsCalculate the MEDIAN for the above example.

The total frequency is ∑f = 20. The median value is themiddle value. There are 20 values so the median is themean of the 10th and 11th values when they are put in order. These occur in the 3rd row of the table.

So the MEDIAN = 2

( Median is at 1 (∑f +1 ) = 1 (20 +1 ) 2 2 = 1 x 21 2 = 10.5th Value )

Averages for Frequency Distributions

Calculate the MODE for the above example.

The mode is the number of goals with the

greatest frequency. In the table the greatest

frequency is 6 so the MODE = 2.

Grouped Frequency Distributions When there is a lot data, or the data is CONTINUOUS,

GROUPED FREQUENCY DISTRIBUTIONS are used. For a grouped frequency distribution with equal class width

intervals, the MODAL CLASS is the group with the LARGEST FREQUENCY.

For a grouped frequency distribution the exact value of the MEAN cannot be calculated because the actual values of the data in each group are not known. We assume that al the data values in a group are located at the MID-POINT of the class and use this to ESTIMATE the MEAN using;ESTIMATED MEAN = ∑fx

∑fwhere ∑fx is the total of all the class mid-point multiplied by their associated frequencyand ∑f is the total frequency

Averages for Grouped Frequency Distributions

Eg Michael measured, correct to the nearest millimetre, the size of the hand span for 25 pupils, here are the results. Work out the ESTIMATED MEAN hand span.

Hand Span Frequencyf

Midpointx

Frequency x Midpointfx

9.0 – 9.910.0 – 10.911.0 – 11.912.0 – 12.913.0 – 13.914.0 – 14.9

224665

9.4510.4511.4512.4513.4514.45

2 X 9.45 = 18.92 X 10.45 = 20.94 X 11.45 = 45.86 X 12.45 = 74.76 X 13.45 = 80.7

5 X 14.45 = 72.25

∑f = 25 ∑fx = 313.25

Averages for Grouped Frequency Distributions

The easy way to do this is to add two more columns to the table. One is for the mid-point of each group and the other is for an estimate of the total hand spanESTIMATED MEAN = ∑fx ∑f = 313.25 25

= 12.53cmWhere:∑fx is the sum of the handspans∑f is the sum of the pupils

Averages for Grouped Frequency Distributions

Find the MODAL CLASS INTERVAL for the above example.

The MODAL CLASS INTERVAL is the group with thegreatest frequency. So the MODAL CLASS INTERVALSare 12.0 – 12.9 and 13.0 – 13.9.

For the above example find the CLASS INTERVAL inwhich the MEDIAN lies.

In this example the total frequency is ∑f = 25. TheMEDIAN value is the middle value. There are 25 values sothe median is the 13th value when they are put in order.This occurs is the 4th row of the table. So the MEDIAN liesin 12.0 – 12.9 CLASS INTERVAL.

Which is the Best Average to Use? Sometimes the mean is the best eg test result, sometimes the mode

is the best eg stock control and sometimes the median is best eg house prices.

Advantages And Disadvantages Of The Three Statistical Averages

Average Advantages Disadvantages

Mean Widely used

Makes use of all the data

May not correspond to an actual value

Affected by extreme values

Mode Represents an actual value

Easy to obtain from diagrams

Does not take account of all values

There may be no mode or many modes

Median Often represents an actual value

Is not affected by extreme values

Not widely used

Not representative of a small group