# S1 Measures of Dispersion The mean, variance and standard deviation

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```					S1 Measures of Dispersion
The mean, variance and
standard deviation
S1 Measures of Dispersion
Objectives:
To be able to find the variance
and standard deviation for
discrete data

To be able to find the variance
and standard deviation for
continuous data
A factory worker counted the numbers of nuts in a
packet. The results are shown in the table
Number    Freq    CF     Calculate P80 and P40
of nuts
6      8        8     P80 = 99/100x80 = 79.2
7      12      20     P80 = 80th term
8      36      56     P80 = 10 nuts
9      18      74     P40 = 99/100x40 = 39.6
10      15      89     P40 = 40th term
11      10      99     P40 = 8 nuts

Calculate the 40th to 80th interpercentile range
40th to 80th interpercentile range = 10 - 8 = 2 nuts
The lengths of a batch of 2000 rods were measured to the nearest
cm. The measurements are summarised below.
Length        Number of     Cumulative
(nearest cm) rods           frequency
Q1=74.5 + 500-250 x 5
738-250
60-64          11              11          Q1=77.06
65-69          49              60
70-74          190                         Q2=79.5+1000-738 x 5
250                  1370-738
75-79          488             738         Q2=81.57
80-84          632             1370
85-89          470             1840        Q3=84.5+1500-1370 x 5
90-94          137             1977                1840-1370
Q3=85.88
95-99          23              2000
By altering the formula slightly can you work out how to find the 3rd
decile (D3) and the 67th percentile (P67)?
D3=74.5 + 600-250 x 5
738-250
D3=78.09

P67=79.5 + 1340-738 x 5
1370-738
P67=84.26
Variance of discrete data
The deviation (difference) of the data (x) from
_
the mean (x) is one way of measuring the
dispersion (spread) of a set of data.
_
Variance = Σ(x – x)²
n
_
Variance = Σx² – Σx ²
n     n
Standard deviation.
As variance is measured in units ² you usually
take the square root. This gives you the
standard deviation.
Standard deviation = √variance

Standard deviation symbol is σ
The marks scored in a test by seven students
are 3,4,6,2,8,8,5. Calculate the variance and
standard deviation.
x     x²                             _
Variance = Σx² – Σx ²
3     9
n     n
4     16
36
6                σ ² = 218 – 36 ² = 4.69
2     4                7     7
8     64
64
σ = √4.69 = 2.17
8
5     25
Σx = 36 Σx² = 218
The variance and standard deviation from a
frequency table
Number of     Freq of
rods (x)     rods (f)     fx      fx²
35            3       105      3675
36           17       612      22032
37           29       1073     39701
38           34       1292     49096
39           26       1014     39546
Σf=109 Σfx=4096 Σfx²=154050

Variance = Σfx² – Σfx ²     Variance = 154050 – 4096 ²
Σf     Σf                   109      109

σ ²=1.19805       σ = 1.109
Variance and standard deviation from a grouped frequency distribution

Height of     Freq
plant (cm)    (f)
0<h≤5         4
5 < h ≤ 10    15
10 < h ≤ 15   5
15 < h ≤ 20   2
20 < h ≤ 25   0
25 < h ≤ 30   1
Total

σ ² = 6487.5 – 285      ² = 128.85802     σ   = √128.85802
27        27                     σ   = 11.35

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 views: 54 posted: 11/25/2011 language: English pages: 10
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