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 = 69 years When Carly’s mum joins the group. Total age = = years The new mean will be: __ X = = 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) = = 1584 Total number of children = = 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. __ X = = = £ Σ f = 25Σ fx = 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. The weighted mean mark = Σ wx ΣwΣw Where w is the weighting given to each value of x. Σw = 100Σwx = 7960 = 7960 = 79.6% 100 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. Σw = 6Σwx = 120 The mean price of a ticket = Σ wx ΣwΣw = 120 = £20 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: (-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: (5 x 9 x 12) = 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) = … The equivalent single rate is 6.96% to 3 sf. Great Marlow School Mathematics Department