Mode, Median and Mean Great Marlow School Mathematics Department

Calculating the mode, median and mean for a grouped frequency distribution of discrete data A local shop sells mobile phones and keeps a record of the daily sales. The table shows these sales. The modal group is 15 – 19 phones. It is impossible to find the exact value of the median when the data has been grouped. It can be estimated by using interpolation. The total frequency Σf = 29 The position of the median is 29 + 1 = 15th value 2 To find this value, use a table with running values. Great Marlow School Mathematics Department

Estimating the median using interpolation Look at the group 10 – 14. The group is 5 numbers wide: 10, 11, 12, 13 and 14. The frequency is 4, so the group has 4 values in it. The 15th value is the 2nd of these 4 values. So the estimate would be 2 of the way through this group. 4 An estimate of the median = 10 + 2 x 5 = 12.5 Great Marlow School Mathematics Department

Estimating the mean Σf = 29 Σfx = 333 It is impossible to find the exact value of the mean when the data has been grouped. It can be estimated by using the mid-values of each group. The mid-values are ½(0 + 4) = 2, ½(5 + 9) = 7 and so on. To work out the mean add two extra columns to your table. Use the mid-values as the x values. Estimate of the mean: __ X = Great Marlow School Mathematics Department

Calculating the mode, median and mean for a grouped frequency distribution of continuous data You cannot give an exact value to continuous data because it is impossible to measure it exactly. Continuous data always has to be given to a chosen degree of accuracy. The groups for continuous data need some adjusting when working out means and medians. Measurements of time, weight height and speed are often given to the nearest unit. The groups for these are usually written like this: Great Marlow School Mathematics Department

This means 30.5 up to, but not including 40.5 You use 30.5 and 40.5 to work out the mid-value You use 30.5 as the beginning of the median calculations. 21 – 30 31 – 40 41 – 50 100 – 200 – 300 – This means 200 up to, but not including 300 You use 200 and 300 to work out the mid-value You use 200 as the beginning of the median calculation Special care is needed with age, if it is counted in completed years. This means 31 up to, but not including 41 You use 31 and 41 to work out the mid-value You use 31 as the beginning of the median calculation 21 – 30 31 – 40 41 – 50 Great Marlow School Mathematics Department

Sometimes algebra is used in a table to show the size of the group. 10 is not included in this group but 20 is included You use 10 and 20 to work out the mid-values You use 10 as the beginning of the median calculations 10 < x 20 10 is included in the group but 20 is not included You still use 10 and 20 to work out the mid-value You still use 10 as the beginning of the median calculations 10 X < 20

Example: The table shows the time in minutes, to the nearest minute, spent by people travelling to work. Estimate the median time spent travelling to work. The median is 44 / 2 = 22nd value An estimate for the median = 14.5 + 12/14 x (29.5 – 14.5) = 27.4 minutes to 1 d.p. Great Marlow School Mathematics Department

Write down the modal group. Exercise 3:4 Question 1 Dillon collects apples from the tree in his garden. He weighed each one and recorded the weight, to the nearest gm, in a table. Σf = Σfx = Write down the modal group. Estimate the mean weight, to the nearest gram, of Dillon’s apples. Estimate the median weight, to the nearest gram, of an apple in Dillon’s garden. Great Marlow School Mathematics Department

Exercise 3:4 Question 1 Dillon collects apples from the tree in his garden. He weighed each one and recorded the weight, to the nearest gm, in a table. Σf = 32 Σfx = 5276 Modal group = 161 – 180 gm __ X = (b) An estimate of the mean = 5276 = 164.875 = 165 gm (c) The median is the = 33/2 = 16.5 = 17th value An estimate of the median =160.5 + 3/9 x (180 – 161) = 166.8 gms Great Marlow School Mathematics Department

Question 2: Carla measures the height of pupils in her class in cm Question 2: Carla measures the height of pupils in her class in cm. The table gives her measurements. Σf = Σfx = For the pupils in Carla’s class: Write down the modal group. Estimate the mean height to the nearest centimetre. Estimate the median height to the nearest centimetre. Great Marlow School Mathematics Department

Question 2: Carla measures the height of pupils in her class in cm Question 2: Carla measures the height of pupils in her class in cm. The table gives her measurements. Σf = 28 Σfx = 3620 110 ≤ x < 120 The modal group = An estimate of the mean = 3620/28 = 129.28 cm = 129.3 cm The median is the 29/2 = 14.5 = 15th value An estimate of the median = 120 + 5/6 x (130 – 120) = 128.3 cm Great Marlow School Mathematics Department