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Averages And Range Homework

We use statistics such as the mean, median and mode to obtain information about a population from our sample set of observed values.


Mean

The mean (or average) of a set of data values is the sum of all of the data values divided by the number of data values.  That is:


Example 1

The marks of seven students in a mathematics test with a maximum possible mark of 20 are given below:
     15     13     18     16     14     17     12

Find the mean of this set of data values.

Solution:

So, the mean mark is 15.


Symbolically, we can set out the solution as follows:

So, the mean mark is 15.


Median

The median of a set of data values is the middle value of the data set when it has been arranged in ascending order.  That is, from the smallest value to the highest value.


Example 2

The marks of nine students in a geography test that had a maximum possible mark of 50 are given below:
     47     35     37     32     38     39     36     34     35

Find the median of this set of data values.

Solution:

Arrange the data values in order from the lowest value to the highest value:

     32     34     35     35     36     37     38     39     47

The fifth data value, 36, is the middle value in this arrangement.


Note:


In general:

If the number of values in the data set is even, then the median is the average of the two middle values.


Example 3

Find the median of the following data set:
     12     18     16     21     10     13     17     19

Solution:

Arrange the data values in order from the lowest value to the highest value:
     10     12     13     16     17     18     19     21

The number of values in the data set is 8, which is even.  So, the median is the average of the two middle values.


Alternative way:

There are 8 values in the data set.

The fourth and fifth scores, 16 and 17, are in the middle.  That is, there is no one middle value.


Note:
  • Half of the values in the data set lie below the median and half lie above the median.
  • The median is the most commonly quoted figure used to measure property prices.  The use of the median avoids the problem of the mean property price which is affected by a few expensive properties that are not representative of the general property market.


Mode

The mode of a set of data values is the value(s) that occurs most often.

The mode has applications in printing.  For example, it is important to print more of the most popular books; because printing different books in equal numbers would cause a shortage of some books and an oversupply of others.

Likewise, the mode has applications in manufacturing.  For example, it is important to manufacture more of the most popular shoes; because manufacturing different shoes in equal numbers would cause a shortage of some shoes and an oversupply of others.


Example 4

Find the mode of the following data set:

     48     44     48     45     42     49     48

Solution:

The mode is 48 since it occurs most often.


Note:
  • It is possible for a set of data values to have more than one mode.
  • If there are two data values that occur most frequently, we say that the set of data values is bimodal.
  • If there is no data value or data values that occur most frequently, we say that the set of data values has no mode.


Analysing Data

The mean, median and mode of a data set are collectively known as measures of central tendency as these three measures focus on where the data is centred or clustered.  To analyse data using the mean, median and mode, we need to use the most appropriate measure of central tendency.  The following points should be remembered:

  • The mean is useful for predicting future results when there are no extreme values in the data set.  However, the impact of extreme values on the mean may be important and should be considered.  E.g. The impact of a stock market crash on average investment returns.
  • The median may be more useful than the mean when there are extreme values in the data set as it is not affected by the extreme values.
  • The mode is useful when the most common item, characteristic or value of a data set is required.

Key Terms

statistics, mean, average, median, mode, measures of central tendency


Mean, Median, Mode and Range

Skills needed:
  • Addition
  • Multiplication
  • Division
  • Data Sets
When you get a big set of data there are all sorts of ways to mathematically describe the data. The term "average" is used a lot with data sets. Mean, median, and mode are all types of averages. Together with range, they help describe the data.

Definitions:

Mean - When people say "average" they usually are talking about the mean. You can figure out the mean by adding up all the numbers in the data and then dividing by the number of numbers. For example, if you have 12 numbers, you add them up and divide by 12. This would give you the mean of the data.

Median - The median is the middle number of the data set. It is exactly like it sounds. To figure out the median you put all the numbers in order (highest to lowest or lowest to highest) and then pick the middle number. If there is an odd number of data points, then you will have just one middle number. If there is an even number of data points, then you need to pick the two middle numbers, add them together, and divide by two. That number will be your median.

Mode - The mode is the number that appears the most. There are a few tricks to remember about mode:

If there are two numbers that appear most often (and the same number of times) then the data has two modes. This is called bimodal. If there are more than 2 then the data would be called multimodal. If all the numbers appear the same number of times, then the data set has no modes.

They all start with the letter M, so it can be hard to remember which is which sometimes. Here are some tricks to help you remember:
  • Mean - Mean is the average. It's also the meanest because it take the most math to figure it out.
  • Median - Median is the middle. They both have a "d" in them.
  • Mode - Mode is the most. They both start with "mo".
Range - Range is the difference between the lowest number and the highest number. Take, for example, math test scores. Let's say your best score all year was a 100 and your worst was a 75. Then the rest of the scores don't matter for range. The range is 100-75=25. The range is 25.

Example problem finding mean, median, mode and range:

Find the mean, median, mode and range of the following data set:

9,4,17,4,7,8,14

Finding the mean:

First add the numbers up: 9+4+17+4+7+8+14 = 63

Then divide 63 by the total number of data points, 7, and you get 9. The mean is 9.

Finding the median:

First put the numbers in order: 4, 4, 7, 8, 9, 14, 17

The middle number is 8. There for the median is 8.

Finding the mode:

Remember the mode is the number that appears the most. It can help to put the numbers in order so we don't miss anything: 4, 4, 7, 8, 9, 14, 17

Four appears twice and the rest of the numbers only appear once. The mode is 4.

Finding the range:

The lowest number is 4. The highest number is 17.

Range = 17 - 4

Range = 13



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