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| Unit 5 ~ Quantitative Techniques for Business | |
| Lesson 6 - 14th March 2000 | |
| Frequency ~ Dispersion | |
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What is Dispersion?
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Download the "Range, 900 metre, Fire" exercise in pdf format (94Kb).
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Dispersion
What is Dispersion?
- how data is dispersed.
- how data is spread over the whole range of data.
- mean, median and mode give an indication of central tendency, dispersion gives an indication of spread.
- measures of spread include: range; mean absolute deviation, variance and standard deviation
- the range is the difference between the highest data value and the lowest data value.
- with a data range such as: 95, 52, 13, 45, 59, 23,
37, the range is
95-13 = 82 - the range does not tell us how the data is spread, but does tell us the extent of the spread.
- a fractile is a statistic that denotes the boundary of a fractional part of the data
- the most commonly used are:
- quartiles
- deciles
- percentiles
there are three quartiles and they divide the data into four equal parts
the lower quartile is one quarter through the data, the upper quartile is three quarters through the data, the middle quartile is two quarters through the data (is halfway) so is calculated in the same way as the median
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Deciles
there are nine deciles and they divide the data into ten equal parts
Percentiles
there are ninety-nine percentiles and they divide the data into one hundred equal parts
Mean Absolute Deviation
here we measure the average deviation from the mean.
consider a very simple set of numbers
1, 2, 3, 4, 5, the mean is 3
Mean Absolute Deviation
| Number | Deviation from the Mean | Absolute Deviation |
| 1 | 3 - 1 = 2 | 2 |
| 2 | 3 - 2 = 1 | 1 |
| 3 | 3 - 3 = 0 | 0 |
| 4 | 3 - 4 = -1 | 1 |
| 5 | 3 - 5 = -2 | 2 |
| Total | 6 |
the sum of the absolute deviations is 6
the total absolute deviation from the mean is 6
the average absolute deviation from the mean is 6 + 5 = 1.2
the mean absolute deviation (MAD for short) from the mean is 1.2
The Variance
we can overcome the problems of "negative deviations" by squaring the deviations, because when two negative numbers are multiplied together the result is a postive number.
The Variance
| Number | Deviation from the Mean | Squared Deviation |
| 1 | 3 - 1 = 2 | 4 |
| 2 | 3 - 2 = 1 | 1 |
| 3 | 3 - 3 = 0 | 0 |
| 4 | 3 - 4 = -1 | 1 |
| 5 | 3 - 5 = -2 | 4 |
| Total | 10 |
the sum of the squared deviations is 10
the total squared deviation from the mean is 10
the average squared deviation from the mean is 10 + 5 = 2
the variance from the mean is 2
the only problem now is units, if it was cm, the variance is now cm²
Standard Deviation
standard deviation is the square root of the variance, as the variance is the sum of the squares of differences, then taking the square root of that sum is logical
standard deviation is the most commonly used measure of dispersion
using the previous example, the standard deviation is the square root of 2 which is equal to 1.41 (two decimal places).
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