1. Business Decision Making
Measures of Dispersion & Skewness
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2. Learning Outcomes Evaluate various measures of dispersion.
By the end of this session, students should be able to:
Evaluate various measures of dispersion.
Understand standard deviation, its significance & interpretation.
Discuss Skewness of a distribution and calculate the coefficient of skewness.
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3. Measures of Dispersion
Obtaining the average value for representation of set of data, logical question evolves – To what extent does this single value represent the whole set?
Measure of Dispersion signifies the degree to which the values (data) in the distribution are ‘spread out’ OR ‘squashed’ together – the degree of spread among those values.
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4. Measures of Dispersion
Range:
It is the difference between the highest and lowest values in a distribution.
Range = Highest Value – Lowest Value
As it ignores all but the two extreme values of a distribution, what has happened in the middle is not accounted for by this measure.
Just like mode, range is only useful when we want a quick idea of the variability in a set of data without having to go to the trouble of doing any calculations. (Morris 2000)
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5. Measures of Dispersion
Quartile:
Overcoming the criticism of range about ignoring the middle values, Quartile is utilised, which looks at the range, but not among all 100% of the data, but only among the central 50% of the distribution. (Morris 2000)
The spread between the values 25% of the way and 75% of the way through a distribution are examined; respectively called the first or lower, quartile Q1 and the third or upper, quartile Q3. What about the Middle one??
They are respectively 1/4 (one quarter) & 3/4 (three quarters) of the way through the distribution.
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6. Measures of Dispersion
Quartile Deviation:
This measure reveals the spread of the both upper (Q3) and lower (Q1) with respect to the Median (M).
As Q3 – Q1 gives us the Interquartile Range (Semi-interquartile range = [(Q3 – Q1)/2]) the differences (Q3 – M) & (M – Q1) would reveal their distances from the Median, thus informing us the idea of the general shape of the distribution.
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7. Measures of Dispersion
Standard Deviation (SD):
It is a “measure that tells us how near, on the whole, the values in the data are to the mean.” (Morris 2000)
An easy example of understanding calculation of SD:
Consider a set of numbers 1, 2, 3, 5, 9. The mean of this set is 4. 1 2 3 5 9
- 3
- 2 4 25 40
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8. Measures of Dispersion
Henceforth, the average squared deviation of each values if 40/5 = 8, which is called the ‘Variance’.
However, for different sets of data units, this may not deliver much meaning. For instance, lengths, areas, percentages, etc!
Therefore, we use the square-root of variance to get the SD; i.e. sq. root of 8 =
Symbolically, we use:
where:
Grouped Data: 8
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9. Measures of Dispersion- Standard Deviation
Recall the height distribution of employees: 9
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10. Measures of Dispersion- Standard Deviation
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11. Measures of Dispersion- Coefficient of Variance
Coefficient of Variance is a statistical measure of the dispersion of data points in a data series around the mean. It is calculated as the ratio of standard Deviation to mean.
It is useful when comparing the degree of variation from on data series to another, even if the means are quite different from each other.
Look at the example:
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12. Measures of Dispersion- Coefficient of Variance
Government statistics on the basic weekly wages of workers in 2 counties of London show following figures:
Can we conclude that county V has wider spread of basic weekly wages?
Solution:
Simple observation of ‘’ may suggest a positive answer. However, using coefficient of variance,  / shows following:
Coefficient of variation of wages in county V = 55/120 = 45.8%
Coefficient of variation of wages in county W = 50/90 = 55.6%
Which shows, in fact, that it is country ‘W’ that has a higher variability in basic weekly wages.
County V
County W
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13. Example for you to workout!
Calculate the standard deviation for following set of distribution for a factory that produces ‘Q’s every day.
Formula:
Output product Q
No. of days
350 – under 360
360 – 370
370 – 380
380 – 390
390 – 400 4 6 5 3
Ans: 13.02 13
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14. Skewness Skewness signifies asymmetry or unequal distribution!
If items in a distribution are equally dispersed on each side of the mean, they are said to be ‘symmetrical’ distribution, as shown in the figure.
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15. Skewness
When distribution is not symmetrically dispersed on each side of the mean, then they are said to be skewed distribution or asymmetric.
There can be 2 skew distribution as shown.
(a) Positively Skewed & (b) Negatively Skewed
Such distribution can have same mean & same standard deviation but differently skewed? 15
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16. Skewness Information that skewness delivers:
- Positive skewness: Expect unusually high values in the distribution.
- Negative skewness: Expect unusually low values in the distribution.
Symmetrical Distribution has its Mean, Median & Mode all on the same point as shown on 13th Slide. (on the dotted line)
For skewed distribution, they lie at different place with Mean on the tail side with median followed by mode at tallest point, for both positive & negative skewed.
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17. Skewness
For most skewed distributions (except those with long tails) following relationship holds:
Mean – Mode = 3 (Mean – Median)
More skewed the distribution, more spread out these measures of location; therefore the amount of spread can be utilised to calculate the skewness of distribution.
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18. Coefficient of Skewness
Usually, the skewness is calculated by;
Pearson’s First Coefficient of Skewness =
As mode not always very easy to ascertain, equivalent formula can be used:
Pearson’s Second Coefficient of Skewness =
Candidates should be careful while subtracting ‘mode’ or ‘median’ from ‘mean’ as (+) or (-) sign indicate Positive Or Negative skewness. 18
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