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Additional worked example
T 3M
c t
:
easures of
entral
endency
Sum
The mean is defined according to the formula X =
N
∑ x = 1034 = 73.86
N
14
The median is found by first calculating the median location according to the formula:
N +1
median l ocation =
2
So for our data,
N + 1 14 + 1 15
median location =
=
=
= 7.5
2
2
2
1.1.2.
∑x
So for our data, X =
1.1.1.
Score
56
59
62
68
73
74
75
78
80
80
81
82
82
84
1034
Rank
1
2
3
4
5
6
7
8
9
10
11
12
13
14
1. Calculate the mean, median and mode for the following set of data:
utorial
The median is found midway between the 7th and 8th value. These are 75 and 78 and so the
median is 76.5
Finally, the mode is the most commonly occurring value. Our data indicates that two values
– 80 and 82 - both occur twice. As such, both are modes and the distribution can be said to
be bimodal.
1.3
2. Calculate the mean, median and mode for the following data:
Scores
1
2
3
4
5
6
7
41
57
65
66
69
81
82
461
Sum
Rank
Begin by ranking the data and calculating the sum of x:
Scores
65
82
41
69
57
66
81
2.1. To calculate the mean, use the formula:
For our data, X =
∑x
N
X =
∑ x = 461 = 65.86
N
7
2.2. To calculate the median, first find the median location. Using the formula:
median location =
N +1
2
The median location is:
N +1 7 +1 8
=
= =4
2
2
2
median location =
So the median is that value corresponding to location 4, which turns out to be 66.
2.3. To calculate the mode, we look for the most frequently occurring value. In this case, each value
occurs only once, and as such there is no mode.
3. Find the mean, median and mode to the following data:
Scores 39 37 37 36 36 34 33 95 11 37
Scores
Sum
11
33
34
36
36
37
37
37
39
95
395
1
2
3
4
5
6
7
8
9
10
Rank
Begin by ranking the data:
Scores
33
34
36
36
37
37
37
39
289
2
3
4
5
6
7
8
9
Rank
3.1. In order to calculate the mean, the outliers are first discarded to produce a trimmed mean:
Sum
Then, the mean is worked out using the formula:
X =
N
So for our data,
∑x
X =
∑ x = 289 = 36.125
N
8
3.2. To work out the median, we first calculate the median location:
median location =
N +1 8 +1 9
=
= = 4.5
2
2
2
The median location is 4.5, which corresponds average of values 36 and 37. So the median is 36.5
3.3. The mode is defined, quite simply, as the most frequently occurring value. In this case, it is 37.
4. Find the mode, median and mean for the following sample:
Scores
5
8
11
9
8
6
8
We begin by ranking the data:
Scores
1
2
3
4
5
6
7
5
6
8
8
8
9
11
55
Sum
Rank
So for our data,
X =
∑x
N
X =
4.1. In order to calculate the mean, the following equation is used:
∑ x = 55 = 7.86
N
7
4.2. The median location is defined as
median location =
N +1
2
So for our data,
median location =
N +1 7 +1 8
=
= =4
2
2
2
Location 4 corresponds to the value 8, so the median is 8.
4.3. The mode corresponds to the value with the highest frequency. In this case, it is 8.