Cambridge International AS & A Level Mathematics: Probability & Statistics 1
PS
8 Three classes of continuous data are given as 0–4, 4–10 and 10–18. The frequency densities of the classes 0– 4 and 10–18 are in the ratio 4:3 and the total frequency of these two classes is 120. Find the least possible frequency of the modal class, given that the modal class is 4–10.
2.2 The mean
E
The mean is referred to more precisely as the arithmetic mean and it is the most commonly known average. The sum of a set of data values can be found from the mean. Suppose, for example, that 12 values have a mean of 7.5 : sum of values sum of values Mean = , so 7.5 = and sum of values = 7.5 × 12 = 90. number of values 12 You will soon be performing calculations involving the mean, so here we introduce notation that is used in place of the word definition used above.
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We use the upper case Greek letter ‘sigma’, written Σ, to represent ‘sum’ and x to represent the mean, where x represents our data values. The notation used for ungrouped and for grouped data are shown on separate rows in the following table. Sum
Data values
Frequency of data values
Number of data values
Sum of data values
Ungrouped
Σ
x
–
n
Σx
x=
Σx n
Grouped
Σ
x
f
Σf
Σxf or Σfx
x=
Σxf Σf
SA M
30
Mean
WORKED EXAMPLE 2.3
TIP
n = Σf for grouped data.
TIP
Σxf = Σfx indicates the sum of the products of each value and its frequency. For example, the sum of five 10s and six 20s is (10 × 5) + (20 × 6) = (5 × 10) + (6 × 20) = 17 Σx We first rearrange the formula x = (10 × 5) + (20 × 6) = (5 × 10) + (6 × 20) = 170.
Five labourers, whose mean mass is 70.2 kg, wish to go to the top of a building in a lift with some cement. Find the greatest mass of cement they can take if the lift has a maximum weight allowance of 500 kg. Answer
Σx = x × n = 70.2 × 5 = 351kg
351 + y = 500 y = 149
n to find the sum of the labourers’ masses.
We now form an equation using y to represent the greatest possible mass of cement.
The greatest mass of cement they can take is 149 kg.
Original material © Cambridge University Press 2018
KEY POINT 2.2 For ungrouped data, Σx . the mean is x = n For grouped data, Σfx Σxf x= or . Σf Σf