Teach GCSE Maths

More about the three More about the three MsMs

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More about More about the three Msthe three Ms

There are 3 Ms: the Mode, the Median and the Mean. e.g. 1 3 5 6 7 8 9

Which of the 3 Ms has no meaning for this data set?

ANS: There is no mode ( most ).

Here there is more than 1 mode.

1 3 3 6 7 9 9

The mode tells us which number appears most often in a data set but some data sets have no

mode and others have several modes.

The median can always be found but if a data set has an even number of values, there is not just one number in the middle.

3 4 6 6 8 9e.g.1

median = 6

We must look at the 2 middle numbers.

e.g.2 1 7 8 10 12 14

The 2 middle numbers are different. The median is the middle ( or average ) of these 2 numbers. = 9median

Sometimes we can find all three Ms but one, or more, does not represent the data well.

Solution:

e.g. Find the modes, medians and means for the following two data sets:

1 1 1 3 4 18 21

1 1 1 3 4 5 6

Set A:

Set B:

Set A Set B

Mode

Median

Mean

1 1

3 3

7 3

The mode and median are not affected by the large values in set A but the mean takes

account of all the numbers so represents the data better.

The three Ms are sometimes all referred to as averages.For example, you may be asked which average is the most suitable to use to compare the salaries of 2 businesses.Here the question wants you to choose between the mode, the median and the mean.

However, the word average is also used to refer just to the mean.

You need to know about the 2 uses of the word average, but don’t worry about it. It will be obvious which is meant. If you are asked to average some numbers, you need to find the mean.

( The median is best as a couple of very big numbers won’t have a large effect on the result )

If there is one unknown number in a data set, we can find it if we know the mean.

e.g.1. If the mean is 7, find the missing value in this data set

4 6 …Solution

:If we replace all the numbers in a data set by the mean, the total stays the same.

7 + 7 + 7 = 21 or 3 7 =The sum of the given numbers = 4

+ 6 =So, the missing number =

The mean is 7:

= 11

2110

21 10

e.g. 2. The mean of 10 numbers is 4·8. Nine of the numbers are

1, 2, 2, 3, 3, 5, 7, 10, 11Find the 10th

number.Solution: The mean is 4·8, so the total is 10

4·8 Total =The sum of the 9 given numbers

So, the missing number = = 4

4448 44

= 1 + 2 + 2 + 3 + 3 + 5 + 7 + 10 + 11 =

48

SUMMARY

Some data sets have no mode or several modes. The median of a data set with an even number of values is the average of the 2 middle values ( in order ).

To find a missing value if the mean is known:• Find the total by multiplying the mean

by the number of values ( including the missing one ).• Add up the known values.• Subtract.

Exercise

1. Find the medians for the following data sets:(a) 3, 4, 7, 7, 8, 8 (b)1, 2, 4, 6, 8, 10(c) 2, 5, 1, 4, 3 (d)2, 5, 3, 5, 4, 6

Solutions:(a) 3, 4, 7, 7, 8, 8 Median = 7

(b)1, 2, 4, 6, 8, 10 2Median

=(4 + 6)1

= 5(c) 2, 5, 1, 4, 3 Reorder

numbers:1, 2, 3, 4, 5

Median = 3(d)2, 5, 3, 5, 4, 6 Reorder

numbers:2, 3, 4, 5, 5, 6

= 4·52

Median =

(4 + 5)1

Exercise

2. Write down 3 numbers that have a median of 5 and a range of 4. ( There is more than one answer. )

3. The mean of three numbers is 6. Two of the numbers are 3 and 7. What is the 3rd number?

Solutions:2. We must have … 5 … to give the median = 5. • One number must be less than, or equal

to, 5 and one greater than, or equal to, 5.

So, the possibilities are:

2, 5, 63, 5, 7 4, 5, 8

1, 5, 55, 5, 9

• To make the range equal to 4, the numbers must differ by 4.

Exercise

3. The mean of three numbers is 6. Two of the numbers are 3 and 7. What is the 3rd number?

Solution:The mean of all three numbers

= 6 So, total = 3 6

The sum of the known numbers = 3 + 7So, 3rd number = 18

10

= 18

= 10

= 8