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S1 Key Points – Measures of Location and Spread
Raw Data
For a set of data values x1 , x2 ,..., xn
The mean, x 
x1  x2  ...  x n

n
x
i
x
or
n
 x
i
 a
n
a
The median is found by arranging the values in order of increasing size. If n is odd,
the median is the 21 n  1 th value. If n is even, the median is halfway between the
1
2
n th value and the following value.
The mode is the value which occurs with the highest frequency. A data set can have
more than one mode if two or more values have the same maximum frequency. A
data set has no mode if all the values have the same frequency.
The range = largest value – smallest value
The interquartile range = upper quartile – lower quartile  Q3  Q1
The variance 
2
1
2
xi  x 2

n
The standard deviation is the square root of variance.
Grouped Data from Frequency Tables
For a data set given in a grouped frequency table, in which the variables takes the
value x1 with frequency f1 , x 2 with frequency f 2 and so on.
The mean, x 
x1 f 1  x2 f 2  ...  xn f n

f 1  f 2  ...  f n
x f
f
i
i
i
The median value can be found by drawing a cumulative frequency graph and
reading off the value of the variable corresponding to a cumulative frequency equal to
half the total frequency.
The modal class is the class with the highest frequency density.
x f
The variance 
f
2
i
i
 x2
2
i
The standard deviation is the square root of variance.
```
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