Transcript Calculations involving the Mean

Calculations involving the Mean
Great Marlow School Mathematics Department
Adding one or more items of data
can change the mean value.
Carly goes to a night club with three of her friends.
Their average age is 17 years and 3 months.
Carly’s mum decides to go, she is 43 year and 6 months.
Find the average age of the group.
• First find the total age of the group including Carly’s mum.
For Carly and her three friends: Mean = total age
4
So the total age = 4 x 17 years and 3 months = 4 x 17.25 = 69 years
• When Carly’s mum joins the group.
Total age = 69 + 43.5 = 112.5 years
• The new mean will be:
__
X =
112.5 = 22. 5 years = 22 years and 6 months
4
Great Marlow School Mathematics Department
Finding the mean of two groups
There are 12 children in Phil’s group. Their mean mark in
a maths test is 76%. In Paul’s group there are only 8
children. Their mean mark is 84%.
Find the overall mean mark for the 20 children.
• Total marks for Phil’s group = 12 x 76 = 912
• Total marks for Paul’s group = 8 x 84 = 672
• Total of all the data values (marks) = 912 + 672 = 1584
• Total number of children = 12 + 8 = 20
• The new mean = 1584 = 79.2%
20
Great Marlow School Mathematics Department
Weighted Mean
It is sometimes important to calculate a mean as a weighted mean.
The table gives the wages and the number of people who earn each
wage in a factory.
Type of Work
Annual
wage (£)
Number of
People
fxx
8000
5
5 x 8000 = 40000
Office worker
11000
4
4 x 11000 = 44000
Glassblower
17000
15
15 x 17000 = 255000
Manager
32000
1
1 x 32000 = 32000
Cleaner
__
X =
 fx
f
Σf = 25
Σfx = 371000
= 371000 = £14840
25
The mean takes into account the number of people getting each wage. It is called a
weighted mean. The numbers of people for each salary are called the weightings.
Great Marlow School Mathematics Department
Weightings can be expressed as percentages
In a Physics examination, there were two papers.
Paper 1 counts for 40% of the final mark and paper 2 counts for
60%. Glenda scores 82 marks out of 100 for Paper 1 and 78 marks
out of 100 for Paper 2. Find her overall percentage mark.
Mark x
82
78
Weighting w
40
60
Σw = 100
The weighted mean mark =
wxx
40 x 82 = 3280
60 x 78 = 4680
Σwx = 7960
Σwx
= 7960 = 79.6%
Σw
100
Where w is the weighting given to each value of x.
Great Marlow School Mathematics Department
Weightings can also be given as ratios
The prices of theatre tickets are £15, £20 and £30. The tickets are
sold in the ratio of 2:3:1 respectively. Find the average price of a
ticket.
Price x Weighting w
15
2
20
3
30
1
Σw = 6
The mean price of a ticket =
wxx
2 x 15 = 30
3 x 20 = 60
1 x 30 = 30
Σwx = 120
Σwx = 120 = £20
Σw
6
Great Marlow School Mathematics Department
Assumed Mean
Kate needs to find the mean of 307, 325, 315, 309, 322
and 318. She guesses what the mean will be.
Kate thinks the mean will be 318.
This guess is called the assumed mean.
Kate finds the difference between the assumed mean and each data value.
These are: -11, 7, -3, -9, 4, 0
She now needs to find the mean of these differences.
(-11) + (7) + (-3) + (-9) + (4) + (0) = -12
Divide by the number of differences -12/6 = -2
Now add the difference to the assumed mean to find the actual mean:
318 + (-2) = 316
The mean of the data values is 316
How can you check this?
Great Marlow School Mathematics Department
Geometric mean
The geometric mean of two numbers is the square root
of there product.
Product means multiply
The geometric mean of 2 and 32 is:
(2 x 32) =
64 = 8
The geometric mean of three numbers is the cube
root of their product.
The geometric mean of 5, 9 and 12 is:
3
(5 x 9 x 12) =
3
540 = 8.1 to 2 sf.
Great Marlow School Mathematics Department
Using the geometric mean
The interest rates for bank accounts may change from
year to year.
The geometric mean can be used to calculate an
equivalent single rate over the two or more years.
A bank pays interest on new accounts at a rate of 10%
for the first year. The rate is 4% for the second year.
To find the value of the account at the end of the two years.
 Multiply the balance by 1.10 (10% plus the original
amount of money)
 This gives the balance at the end if the first year.
 Multiply the balance at the end of the first year by 1.04
 This gives the balance at the end of the second year.
Great Marlow School Mathematics Department
Geometric mean continued…
The geometric mean of 1.10 and 1.04 is
(1.10 x 1.04) = 1.069579…
The equivalent single rate is 6.96% to 3 sf.
Great Marlow School Mathematics Department