,
μ
is the mean of the population and N
is the number of values in the population.5.5 Measures of Central Tendency
Mean – the sum of all values divided by the number of values, also known as average. Mean takes different notation depending on whether you're discussing a sample or a population.
,
μ
is the mean of the population and N
is the number of values in the population.
,
is the mean of the sample and n
is the number of values in the sample.
Median – the value where there is an equal number of values above and below it when ranked sequentially. If there is an odd number of values, then the middle value is the median. If there is an even number of values, then take the mean of the two values closest to the middle.
Mode – the value that occurs the most often. It is possible for a data set to have no mode, or several. Generally we wouldn't consider more than 2 modes at a time.
Outlier – a value distant from the majority of the values.
Ex 1 Determine the mean, median and mode for the data set.
Class Marks: 54, 80, 12, 61, 73, 69, 92, 81, 80, 61, 75, 74, 15, 44, 91, 63, 50, 84
Uses of each measure:
Mean – used the most often, provides a good basis for the middle of the data set.
Median – similar to mean, but outliers have no effect on the median.
Mode – the mode doesn't really measure the middle as much as the other two.
Weighted Mean – used when certain data points are more important than others. I am using a weighted mean to calculate your report card marks.
My plan for your marks this year:
Test 1 weight – 1
Test 2 weight – 3
Test 3 weight – 5
Test 4 weight – 7
Test 5 weight – 4
Test 6 weight – 5
Test 7 weight – 9
To
calculate your grade take your score times the weight for every
entry, then divide by the sum of the weightings. The formula for
weighted mean is
.
Grouped Data
When data is grouped into intervals, we can approximate the mean by using the midpoint of the interval for all the values within the interval.
The
formula looks like:
,
in short the sum of each frequency times its respective midpoint
divided by the sum of the frequencies.
could be used here instead of μ.
Also note the approximate sign is always used because we are using
the midpoint to , you guessed it, approximate the values.
Ex 2 Calculate the mean and median for the number of hours of TV watched in this survey
|
Number of Hours of TV Watched |
Number of children |
|
0-1 |
1 |
|
1-2 |
3 |
|
2-3 |
7 |
|
3-4 |
3 |
|
4-5 |
2 |
|
5-6 |
1 |
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