1. Measures of variation (dispersion) [مقاييس التشتت]
Formula:
For ungrouped data population variance:
2. For ungrouped data sample variance

2. Measures of variation (dispersion) [مقاييس التشتت]
Standard Deviation (S)
1. It is the square root of variance.
2. Most commonly used measure of variance.
3. Shows variation about mean.
4. It used to compare between more than one data set when the means are equal, the best one is the minimum.

3. Measures of variation (dispersion) [مقاييس التشتت]
Example
Calculate variance and standard deviation
Year of graduation
No. of Students
2004 4 -2
2005 6
2006 5 -1 1
2007 8 2
2008 7
Total 30 10

4. Measures of variation (dispersion) [مقاييس التشتت]
Solution: Variance
Standard Deviation
Interpretation: The observations fall 1.58 units from the mean.

5. Measures of variation (dispersion) [مقاييس التشتت]
Variance and standard deviation for grouped data
Formula

6. Measures of variation (dispersion) [مقاييس التشتت]
Example: the table below shows the temperature of a sample of 50 cities taken at the same time on a certain day; determine the mean and standard deviation of the sample.

7. Measures of variation (dispersion) [مقاييس التشتت]
Temp.
Cumulative F.
Midpoint
10-14 10 12
144
120
1440
15-19 22 17
289
204
3468
20-24 18 40
484
396
8712
25-29 6 46 27
729
162
4374
30-34 4 50 32
1024
128
4096
Total
1010
22090

8. Measures of variation (dispersion) [مقاييس التشتت]

9. Measures of variation (dispersion) [مقاييس التشتت]
Coefficient of Variation (C.V) (معامل الاختلاف)
1. It is the main important application of the mean and standard deviation.
2. Measures relative variation and always in percentage (%).
3. Can be used to compare two or more data sets measured in different units.
4. Can be used widely in chemistry and engineering science.
5. The variable with smaller C.V is less dispersed than others so it is the better.

10. Measures of variation (dispersion) [مقاييس التشتت]
Formula:
Coefficient of Variation
Example: Suppose that technician A completes 40 analysis daily with standard deviation of 5, technician B completes 160 analysis per day with standard deviation of 15.
Which employee shows less variability or better?

11. Measures of variation (dispersion) [مقاييس التشتت]
Sol.
Employee B is better than A because he have the less variation.

12. Measures of variation (dispersion) [مقاييس التشتت]
Interquartile Range [IQR] (المدى الربيعي)
There are three quartiles Q1, Q2, Q3
1. Q1 is a 25% of sorted data.
2. Q2 is a 50% of sorted data or median.
3. Q3 is a 75% of sorted data.

13. Measures of variation (dispersion) [مقاييس التشتت]
Formulas
Q2 the same as median formula.

14. Measures of variation (dispersion) [مقاييس التشتت]
Example: you have the frequency table:
Calculate Q1, Q2, Q3 and interquartile range.

15. Measures of variation (dispersion) [مقاييس التشتت]
Sol.
Step (3) the first quartile class is [ ]

16. Measures of variation (dispersion) [مقاييس التشتت]
Sol. Q2
Step (3) the median quartile class is [ ]

17. Measures of variation (dispersion) [مقاييس التشتت]
Sol. Q3
Step (3) the third quartile class is [ ]
Interquartile range = Q3 - Q1 = =11.89.

18. Box plot
A box plot is a descriptive statistics and it is a convenient way of graphically depicting groups of numerical data through their quartiles.
Example: Plot a box plot for {7, 4, 3, 5, 6, 8, 10, 1}

19. Box plot Solution: Sort data as: {1, 3, 4, 5, 6, 7, 8, 10}.
Minimum value is 1, maximum value is 10.
Calculate Q1, Q2, Q3 as:

20. Box plot
Q2 and Q3

21. Box plot

22. Note: The above plotting done by computer using software R as:
Box plot
Note: The above plotting done by computer using software R as:
> A <- c(7, 4, 3, 5, 6, 8, 10 ,1)
> quantile(A)
0% % % % %