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Measures of Variability (Range, SIQR, Standard Deviation)

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Measures of Variability (Range, SIQR, Standard Deviation)

Another important feature of a frequency distribution (in addition to its overall shape and its central tendency) is its variability.

Are the scores all very similar to each other?

Or are they quite different from each other.

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Range

Range - the simplest measure of variability.

Range - is the difference between the highest and the lowest scores.

In most applications, it is best to report the actual highest and lowest scores, as opposed to just the range.

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The range is easy to understand and useful in a lot of situations, but it can be somewhat misleading when there are outliers.

For example, a set of exam scores might range from 90 to 20, making it seem as though there was a high degree of variability.

In fact, all the scores might have been between 80 and 90, except for the one score of 20.

In our example the range is 98 – 37 = 61.

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King Jimens took 7 math tests in one marking period. What is the range of his test scores?89, 73, 84, 91, 87, 77, 94

Least-greatest73, 77, 84, 87, 89, 91, 94

highest - lowest 94 - 73 = 21Range = 21

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SIQR – Semi-Interquartile Range

The semi-interquartile range of a distribution is half the difference between the upper and lower quartiles, or half the interquartile range.

The median divides a set of ordered data into two halves.

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SIQR – Semi-Interquartile Range

The lower quartile is the middle number between the smallest number and the median of the data set.

The second quartile is the middle observation, also called the median of the data.

The third quartile can be measured as the middle value between the median and highest values of the data set.

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SIQR – Semi-Interquartile Range

-Q2 (the middle quartile) is the median. -Q1 (the lower quartile) is the median of

the numbers to the left of, or below Q2. -Q3 (the upper quartile) is the median of

the numbers to the right of, or above Q2.

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Quartiles

Example:12, 14, 16, 18, 20, 22, 24, 26, 28, 30,

32Find the lower, middle and upper

quartiles of the data above.Since the data is already in ascending

order, identify the median.12, 14, 16, 18, 20, 22, 24, 26, 28, 30,

3222 is the median, therefore, Q2= 22

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12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32

The median of the numbers to the left of Q2:

12, 14, 16, 18, 2016 is the median, therefore, Q1 = 16The median of the numbers to the

right of Q2: 24, 26, 28, 30, 3228 is the median, therefore, Q3 = 28

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Interquartile Range

The interquartile range of a distribution is the difference between the upper and lower quartiles.

That is, interquartile range = Q3 – Q1

Therefore using the example above, the interquartile range is:

Interquartile range = Q3 – Q1

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Since,Q3 = 28Q1 = 16

Interquartile range= 28 – 16= 12

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Semi-Interquartile Range

The semi-interquartile range of a distribution is half the difference between the upper and lower quartiles, or half the interquartile range.

Therefore, from the example above, it was determined that the interquartile range = 12.

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Therefore,          semi-interquartile range

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The Standard Deviation

The standard deviation (abbreviated SD or s) is, roughly, the average amount by which the scores differ from the mean.

Consider the following scores: 45, 55, 50, 53, 47. Their mean is 50, and they differ from the mean by 5, 5, 0, 3, and 3. So on average, they differ from the mean by a shade over 3.

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5.3 The Standard Deviation (cont..)

Now consider another set of scores: 20, 80, 50, 38, 62. Their mean is also 50, but they differ from the mean by 30, 30, 0, 12, and 12.

So on average, they differ from the mean by about 17.

These two sets of scores have the same means, but they have quite different standard deviations.

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5.3 The Standard Deviation (cont..)

The standard deviation is not just the mean absolute difference between the scores and the mean. It is just a bit more complicated. Here is how to compute it.

1)Find the mean. 2)Subtract the mean from each score (or each score

from the mean; it does not matter).3)Square each of these differences. 4)Find the mean of these squared differences. 5) Find

the square root of this mean.

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600mm, 470mm, 170mm, 430mm and 300mm

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Find the mean

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Yun lang po!