Analysing data for OCR A-level Maths
This page covers the following topics:
1. Central tendency
2. Standard deviation
Data can be made easier to interpret be calculating different averages, which are called measures of central tendency. The three main measures of central tendency are the mean, the mode and the median. The mean can be found by calculating the sum of the values of the data and dividing by how many values there are. The median can be found by taking the middle value when all the values are put into order from smallest to largest. When there are an even number of values in the set of data, there will be two middle values, and thus the median can be found by calculating the mean of those two values. The mode of a set of values is the value that appears most often. If the values are separated into classes, the modal class is the most common one.
Standard deviation is a measure of spread which gives a quantitative measure of how much the values of a set differ from their mean value. The difference between each value and the mean is found and is squared. These squared differences are then added together and divided by the number of values in the set. Taking the square root of this gives the standard deviation.
Find the median for the following set of values: 52, 68, 59, 61, 70, 55, 72.
We arrange the set of values from smallest to largest: 52, 55, 59, 61, 68, 70, 72. The median is the middle value, therefore the median is 61.
Can the standard deviation of a set of values be calculated using only the number of values in the set and their mean?
The standard deviation cannot be calculated using only these two values, since the difference must be found between each value and the mean. Thus, the values must be known for the standard deviation to be calculated.
Use the given table to calculate the standard deviation for the set of values.
standard deviation = √(((9 + 0 + 1 + 1 + 9) ÷ 5) = 2
The heights of 10 students are recorded as follows: 167, 158, 163, 165, 173, 169, 171, 164, 169, 170. Calculate the mean of their heights. All measurements were taken in centimetres.
Mean = (167 + 158 + 163 + 165 + 173 + 169 + 171 + 164 + 169 + 170)/10 = 166.9 cm.
The standard deviation of a set of values is found to be 2.3. Given that the sum of the squared difference between each value and their mean is 42, calculate the number of values in the set.
2.3 = √(42/n), therefore n = 42/2.3² = 8 (rounded to an integer).
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