Chapter 16 Measures Of Dispersion

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Measures of Dispersion Contents 3.1 Range and Inter-quartile Range 3.2 Box-and-whisker Diagrams 3.3 Standard Deviation 3.4 Comparing the Dispersions of Different Sets of Data 3.5 Effects on the Dispersion with Change in Data

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Measures of Dispersion

3.1 Range and Inter-quartile Range A. Introduction to Dispersion Mean, the median and the mode are measures of the central tendency of a set of data. But such measurements cannot tell us the dispersion of the data. Dispersion is the statistical name for the spread or variability of data Consider the following two sets of data which shows the heights (in cm) of the team members in two different teams. Team A: 160, 161, 162, 162, 163, 164, 165, 167, 171, 175 Team B: 154, 156, 158, 159, 162, 164, 166, 172, 174, 185 Content

Both teams have the same mean height 165 cm but the distribution of the heights of the team members in Team B is more spread out.

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Measures of Dispersion

3.1 Range and Inter-quartile Range B. Range The range is a simple measure of the dispersion of a set of data. For ungrouped data, the range is the difference between the largest value and the smallest value of the set of data. Range = largest value – smallest value

For a set of grouped data, the range is the difference between the highest class boundary and the lowest class boundary. Content

Range = highest class boundary – lowest class boundary

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Measures of Dispersion

3.1 Range and Inter-quartile Range C. Inter-quartile Range When the data is arranged in ascending order of magnitude, the quartiles divide the data into four parts. There are a total of three quartiles which are usually denoted by Q1, Q2 and Q3.

Content

The inter-quartile range is defined as the difference between the upper quartile and the lower quartile of a set of data. Inter-quartile range = Q3 – Q1

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Measures of Dispersion

3.2 Box-and-whisker Diagrams A box-and-whisker diagram illustrates the spread of a set of data. It provides a graphical summary of the set of data by showing the quartiles and the extreme values of the data.

The difference between the two end-points of the line (represented by the highest and lowest marks) is the range. The length of the box is the inter-quartile range.

Content

Fig. 3.16

From the above diagram, we know that the range of the data is 22 and the inter-quartile range is 9.

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Measures of Dispersion

3.3 Standard Deviation A. Standard Deviation for Ungrouped Data For a set of ungrouped data x1, x2, …, xn, ( x1 − x) 2 + ( x2 − x) 2 + ⋅ ⋅ ⋅ + ( xn − x) 2 Standard deviation σ = n n

=

∑ f i ( x1 − x) 2

i =1

n

where x is the mean and n is the total number of data.

Content

Notes: 1. Two sets of data may have the same mean but different standard deviations. 2. The larger the standard deviation, the more spread out the data is.

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Measures of Dispersion

3.3 Standing Deviation B. Standard Deviation for Grouped Data For a set of grouped, we have to consider the frequency of each datum.

Standard deviation σ =

f1 ( x1 − x ) 2 + f 2 ( x2 − x) 2 + ⋅ ⋅ ⋅ + f n ( xn − x) 2 f1 + f 2 + ⋅ ⋅ ⋅ + f n n

=

∑ f i ( x1 − x) 2

i =1

n

∑ fi

i =1

Content

where f i is the frequency of the ith group of data, x is the mean and n is the total number of data.

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Measures of Dispersion

3.3 Standing Deviation C. Finding Standard Deviation by a Calculator Consider the following data: 40 , 36, 47, 53, 56. The following steps demonstrates how to use a calculator to find the mean and the standard deviation of the data. Step 1:

Set the function mode of the calculator to standard deviation ‘SD’ by pressing mode mode 1. then clear all the previous data in the ‘SD’ mode by pressing SHIFT CLR 1 EXE.

Step 2:

Press the following keys n sequence: 40 DT 36 DT 47 DT 53 DT 56 DT

Content

Step 3:

Press SHIFT S-VAR 1 EXE, then we can obtain the mean = 46.4. Press SHIFT S-VAR 2 EXE then we can obtain the standard deviation = 7.55.

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Measures of Dispersion

3.4 Comparing the Dispersions of Different Sets of Data Measure of dispersion

Content

Advantage

Disadvantage

1. Range

Only two data are involved, so it is the easiest one to calculate.

Only extreme values are considered which may give a misleading impression of the dispersion.

2. inter-quartile range

It only focuses on the middle 50% of data, thus avoiding the influence by extreme values.

It cannot show the dispersion of the whole group of data.

3. Standard deviation

It takes all the data into account.

It is difficult to compute without using a calculator. Table 3.34

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Measures of Dispersion

3.5 Effects on the Dispersion with Change in Data A. Removal of a Certain Item from the Data

If the greatest or the least value (assuming both are unique) in a data set is removed, then (1)

the range will decrease;

(2)

the standard deviation will decrease as the data spread less widely;

(3)

the inter-quartile range may increase or decrease.

Content

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Measures of Dispersion

3.5 Effects on the Dispersion with Change in Data B. Adding a Common Constant to the Whole Set of Data If a constant k is added to every datum in a set of data, then the following measures of dispersion (1)

the range,

(2)

the inter-quartile range and

(3)

the standard deviation

will not change.

Content

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Measures of Dispersion

3.5 Effects on the Dispersion with Change in Data C. Multiplying the Whole Set of Data by a Constant

The range, the inter-quartile range and the standard deviation will be k time the original values if the whole set of data is multiplied by a positive constant k.

Content

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Measures of Dispersion

3.5 Effects on the Dispersion with Change in Data D. Insertion of Zero in the Data Set In general, the zero value is the smallest one in a set of data. So, it is similar to the case we studied in Section A. But now, we insert the smallest one into the set of data.

In general, statistical data is non-negative, for example, height, weight, score, etc

If a zero value is inserted in a positive data set, then

• the range will increase; Content

• the standard deviation will increase as the data is spread more widely; • the inter-quartile range may increase of decrease.

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