1. CHAPTER 36 Averages and Range

2. 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.

3. 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 frequency and ∑f is the total frequency

4. Averages for Frequency Distributions Eg 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) 012345012345 456302456302 4 x 0 = 0 5 x 1 = 5 6 x 2 = 12 3 x 3 = 9 0 x 4 = 0 2 x 5 = 10 ∑f = 20∑fx = 36

5. 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 goals Where: ∑fx is the sum of the goals ∑f is the sum of the matches

6. Averages for Frequency Distributions Calculate the MEDIAN for the above example. The total frequency is ∑f = 20. The median value is the middle value. There are 20 values so the median is the mean 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 )

7. 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.

8. 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 ∑f where ∑fx is the total of all the class mid-point multiplied by their associated frequency and ∑f is the total frequency

9. 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 SpanFrequency f Midpoint x Frequency x Midpoint fx 9.0 – 9.9 10.0 – 10.9 11.0 – 11.9 12.0 – 12.9 13.0 – 13.9 14.0 – 14.9 224665224665 9.45 10.45 11.45 12.45 13.45 14.45 2 X 9.45 = 18.9 2 X 10.45 = 20.9 4 X 11.45 = 45.8 6 X 12.45 = 74.7 6 X 13.45 = 80.7 5 X 14.45 = 72.25 ∑f = 25∑fx = 313.25

10. 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 span ESTIMATED MEAN = ∑fx ∑f = 313.25 25 = 12.53cm Where: ∑fx is the sum of the handspans ∑f is the sum of the pupils

11. Averages for Grouped Frequency Distributions Find the MODAL CLASS INTERVAL for the above example. The MODAL CLASS INTERVAL is the group with the greatest frequency. So the MODAL CLASS INTERVALS are 12.0 – 12.9 and 13.0 – 13.9. For the above example find the CLASS INTERVAL in which the MEDIAN lies. In this example the total frequency is ∑f = 25. The MEDIAN value is the middle value. There are 25 values so the median is the 13th value when they are put in order. This occurs is the 4th row of the table. So the MEDIAN lies in 12.0 – 12.9 CLASS INTERVAL.

12. 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 AverageAdvantagesDisadvantages MeanWidely used Makes use of all the data May not correspond to an actual value Affected by extreme values ModeRepresents an actual value Easy to obtain from diagrams Does not take account of all values There may be no mode or many modes MedianOften represents an actual value Is not affected by extreme values Not widely used Not representative of a small group