Primary mathematics/Average, median, and mode

From Wikiversity

Jump to: navigation, search
< Wikiversity < Wikiversity:School of Mathematics < School of Mathematics:Primary School Mathematics


Contents

[edit] Introduction

There are three primary measures of central tendency, and a couple less often used measures, which each, in their own way, tell us what a typical value is for a set of data.

[edit] Basic Measures

[edit] Mode

The mode is simply the number which occurs most often in a set of numbers. For example, if there are seven 12-year olds in a class, ten 13-year olds, and four 14-year olds, the mode is 13, since there are more 13 year olds than any other age. In elections, the mode is often called the plurality, and the candidate who gets the most votes wins, even if they don't get the majority (over half) of the votes.

[edit] Median

The median is the middle value of a set of values. For example, if students scored 81, 84, and 93 on a test; we select the middle value of 84 as the median.

If you have an even number of values, the average of the two middle values is used as the median. For example, the median of 81, 84, 86, and 93 is 85, since that's midway between 84 and 86, the two middle values.

[edit] Average

The straight average, or arithmetic mean, is the sum of all values divided by the number of values. For example, if students scored 81, 84, and 93 on a test, the average is (81+84+93)/3 or 86.

[edit] Weighted average

The weighted average or weighted mean, is similar to the straight average, with one exception. When totaling the individual values, each is multiplied by a weighting factor, and the total is then divided by the sum of all the weighting factors. These weighting factors allow us to count some values as "more important" in finding the final value than others.

[edit] Example

Let's say we had two school classes, one with 20 students, and one with 30 students. The grades in each class on a particular test were:

Morning class = 62, 67, 71, 74, 76, 77, 78, 79, 79, 80, 80, 81, 81, 82, 83, 84, 86, 89, 93, 98
Afternoon class = 81, 82, 83, 84, 85, 86, 87, 87, 88, 88, 89, 89, 89, 90, 90, 90, 90, 91, 91, 91, 92, 92, 93, 93, 94, 95, 96, 97, 98, 99

The straight average for the morning class is 80% and the straight average of the afternoon class is 90%. If we were to find a straight average of 80% and 90%, we would get 85% for the mean of the two class averages. However, this is not the average of all the students' grades. To find that, you would need to total all the grades and divide by the total number of students:

\frac{4300\%}{50} = 86\%

Or, you could find the weighted average of the two class means already calculated, using the number of students in each class as the weighting factor:

\frac{(20)80\% + (30)90\%}{20 + 30} = 86\%

Note that if we no longer had the individual students' grades, but only had the class averages and the number of students in each class, we could still find the mean of all the students grades, in this way, by finding the weighted mean of the two class averages.

[edit] Geometric mean

The geometric mean is a number midway between two values by multiplication, rather than by addition. For example, the geometric mean of 3 and 12 is 6, because you multiply 3 by the same value (2, in this case) to get 6 as you must multiply by 6 to get 12. The mathematical formula for finding the geometric mean of two values is:

\sqrt{AB}

Where:

A = one value
B = the other value

So, in our case:

\sqrt{3(12)} = \sqrt{36} = 6

Note the new notation used to show multiplication. We now can omit the multiplication sign and show simply AB to mean A×B. However, when using numbers, 312 would be confusing, so we put parenthesis around at least one of the numbers to make it clear.

Also, notice that the geometric mean can only be found between two values, not using three or more.


[edit] Advanced Measures

In the Advanced Measures unit, all measures will be referred to as: x_i\,\!,and the mean will be referred to as: \bar x\,\!.

In x_i\,\!, the {}_i\,\! just differentiates the different values of x\,\!, which there could be 10, 20, 30, as many as you want or are given by the problem. Also used in Advanced measures is the sum symbol, which looks like this: \sum\,\!. Always, there will be a number on top of it, below it, to the right of it, and sometimes to the left of it, like so: \sum_{i=1}^nx_i\,\!. All that that complicated symbol means is: "The sum (\sum\,\!) of x_i\,\! (\sum x_i\,\!) with i\,\! from 1 (\sum_{i=1}x_i\,\!) to n, which total, makes \sum_{i=1}^nx_i\,\!.

Sounds complicated, right? Wrong. All that long jumble means is that you add up all the values you have. Now, to get the mean, all you have to do is divide that total by however many numbers you have. So you get:

\bar x=\frac{\sum_{i=1}^nx_i}{n}\,\!

If there is a frequency, then:

\bar x=\frac{\sum_{i=1}^nf_ix_i}{n}\,\!

Now, you can start making a table of all known statistics:

(not necessary)
Index
i\,\!		 Measure
x_i\,\!		 Frequency
f_i\,\!		 f_i x_i\,\!
n\,\!= \sum_{i=1}^nf_ix_i\,\!=

Using \sum_{i=1}^nf_ix_i\,\! and n\,\!, you can find \bar x\,\! with the above formula, and add that to the chart:

Measure
x_i\,\!		 Frequency
f_i\,\!		 f_i x_i\,\! \bar x\,\!
n\,\!= \sum_{i=1}^nf_ix_i\,\!=

So, lets put some measures in and see if you can figure it out:
Exercise 1
Exercise 2
Exercise 3

[edit] Mean Deviation

The mean deviation is a measure that increases as the scores are farther away from the mean. It is mainly used to tell how dispersed the grades are: whether the mean value is every single value, or whether they jump around a lot and it somehow came to the mean. The formula for the mean variation is:

Mean Variation = \frac{\sum_{i=1}^n\left\vert x_i-\bar x\right\vert}{n}\,\!

(with frequency) \frac{\sum_{i=1}^nf_i\left\vert x_i-\bar x\right\vert}{n}\,\!

So, now we need to add new columns to our chart:

x_i-\bar x\,\! \left\vert x_i-\bar x\right\vert\,\! Only with frequencies
f_i\left\vert x_i-\bar x\right\vert\,\!
Only without frequencies
\sum_{i=1}^n\left\vert x_i-\bar x\right\vert\,\!=		 Only with frequencies
\sum_{i=1}^nf_i\left\vert x_i-\bar x\right\vert\,\!=

To Be Continued

Retrieved from "http://en.wikiversity.org/wiki/Primary_mathematics/Average,_median,_and_mode"