. In our example above:|
Section 4 Lesson material |
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(what you should know what to do ) after completing this section you should be able to:
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Now that we have seen how to picture data, we will explore methods of measuring characteristics of data. The measure we first look at is a measure of central tendency. This is a value at the center or middle of a data set.
Consider the following example where we introduce the mean, median, mode and midrange. Here is some data
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10 |
11 |
12 |
12 |
15 |
17 |
21 |
22 |
23 |
27 |
The mean (or arithmetic mean) is the average of these data points. To calculate the mean you simply add the data points and divide by the number of data points. The mean is denoted by . In our example above:
Sum of data points: 10+11+12+12+15+17+21+22+23+27 = 170
Number of data points = 10
Average = 170/10 = 17
The median is the middle value when the scores are arranged in order of increasing (or decreasing) magnitude To calculate the median follow this rule:
NOTE: TO APPLY THE RULES ABOVE THE LISTS MUST BE SORTED!
In our example above: 15 and 17 are the middle numbers. So the median is (15+17)/2 = 16.
The mode of the data set is the score that occurs most frequently. When two scores occur with the same greatest frequency, each one is a mode and the data is bimodal. If more than two scores occur with the same greatest frequency, each is a mode and the data is multimodal. When all scores occur just once there is no mode. The mode is denoted by M
The value 12 in the above dataset occurs most frequently and is therefore the mode.
The midrange is simply (low value + high value)/2. In our example above this is (10+ 27)/2 = 37/2 = 18.5
Mathematicians like to have symbols to represent complicated calculations. Here are some we will use throughout the course:
denotes the summation of a group of values (this means add them all up)
x denotes the variable, usually used to represent the individual data values
n represents the number of values in a sample
N represents the number of values in a population
is the mean of a sample
is the mean of a population
ANOTHER EXAMPLE USING THE NOTATION ABOVE:
Consider the data given below:
|
10 |
12 |
13 |
14 |
489 |
|
15 |
13 |
8 |
14 |
12 |
|
11 |
12 |
13 |
16 |
9 |
Here are the calculations:
40.1
Median - first sort the data
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8 |
9 |
10 |
11 |
12 |
|
12 |
12 |
13 |
13 |
13 |
|
14 |
14 |
15 |
16 |
489 |
Now since the data is odd choose the middle value (there are 15 values - this value #8) which is 13
The data is bimodal with modes of 12 and 13
The midrange is (8+489)/2 = 248.5
WHAT DO YOU THINK ABOUT THIS???
It seems in the above example that the value 489 really skews some of the values - in particular the mean and the midrange. This illustrates the point that both the mean and the midrange are affected by extreme values. The median is not affected and neither is the mode.
This might lead you to believe that the median is a better measure of central tendency than the mean. This is not really true - most statisticians believe that the mean is a better measure (due to the fact that means from samples from a populations are more consistent in general than the median), but hopefully this example shows you that you need to be careful with extreme values
Read about skewness and weighted means in the textbook
Your calculator can do these calculations quickly - if you have a TI-83
Click here to see how to do means, medians from a set of data
Click here to see how to do means, medians from a frequency table
For more discussion on measures of central tendency see these sites:
These links generate descriptive statistics on-line
The data applet (java)
For more info on measure of central tendency see the Exploring data site