1. Averages and Variation
Chapter 3
Averages and Variation
Understanding Basic Statistics
Fifth Edition
By Brase and Brase
Prepared by Jon Booze

2. Measures of Central Tendency
Average – a measure of the center value or central tendency of a distribution of values.
Three types of average:
Mode
Median
Mean

3. Mode The mode is the most frequently occurring value in a data set.
Example: Sixteen students are asked how many college math classes they have completed.
{0, 3, 2, 2, 1, 1, 0, 5, 1, , 0, 2, 2, 7, 1, 3}
The mode is 1.

4. Median Finding the median:
1). Order the data from smallest to largest.
2). For an odd number of data values:
Median = Middle data value
3). For an even number of data values:

5. Median Find the median of the following data set.
{ 4, 6, 6, 7, 9, 12, 18, 19}
a). 6 b). 7 c). 8 d). 9

6. Median Find the median of the following data set.
{4, 6, 6, 7, 9, 12, 18, 19}
a). 6 b). 7 c). 8 d). 9

7. Mean
Sample mean Population mean

8. Mean Sample mean Population mean
Find the mean of the following data set.
{3, 8, 5, 4, 8, 4, 10}
a). 8 b). 6.5 c). 6 d). 7

9. Mean Sample mean Population mean
Find the mean of the following data set.
{3, 8, 5, 4, 8, 4, 10}
a). 8 b). 6.5 c). 6 d). 7

10. Trimmed Mean
Order the data and remove k% of the data values from the bottom and top.
5% and 10% trimmed means are common.
Then compute the mean with the remaining data values.

11. Resistant Measures of Central Tendency
A resistant measure will not be affected by extreme values in the data set.
The mean is not resistant to extreme values.
The median is resistant to extreme values.
A trimmed mean is also resistant.

12. Critical Thinking
Four levels of data – nominal, ordinal, interval, ratio (Chapter 1)
Mode – can be used with all four levels.
Median – may be used with ordinal, interval, of ratio level.
Mean – may be used with interval or ratio level.

13. Critical Thinking
Mound-shaped data – values of mean, median and mode are nearly equal.

14. Critical Thinking
Skewed-left data – mean < median < mode.

15. Critical Thinking
Skewed-right data – mean > median > mode.

16. Weighted Average
At times, we may need to assign more importance (weight) to some of the data values.
x is a data value.
w is the weight assigned to that value.

17. Measures of Variation Three measures of variation: range variance
standard deviation
Range = Largest value – smallest value
Only two data values are used in the computation, so much of the information in the data is lost.

18. Sample Variance and Standard Deviation
Sample Variance Sample Standard Deviation
Find the standard deviation of the data set.
{2,4,6}
a). 2 b). 3 c). 4 d). 3.67

19. Sample Variance and Standard Deviation
Sample Variance Sample Standard Deviation
Find the standard deviation of the data set.
{2,4,6}
a). 2 b). 3 c). 4 d). 3.67

20. Population Variance and Standard Deviation
Population Variance Population Standard
Deviation

21. The Coefficient of Variation
For Samples For Populations

22. Chebyshev’s Theorem

23. Chebyshev’s Theorem

24. Critical Thinking
Standard deviation or variance, along with the mean, gives a better picture of the data distribution than the mean alone.
Chebyshev’s theorem works for all kinds of data distribution.
Data values beyond 2.5 standard deviations from the mean may be considered as outliers.

25. Percentiles and Quartiles
For whole numbers P, 1 ≤ P ≤ 99, the Pth percentile of a distribution is a value such that P% of the data fall below it, and (100-P)% of the data fall at or above it.
Q1 = 25th Percentile
Q2 = 50th Percentile = The Median
Q3 = 75th Percentile

26. Quartiles and Interquartile Range (IQR)

27. Computing Quartiles

28. Five Number Summary Minimum, Q1, Median, Q3, Maximum
A listing of the following statistics:
Minimum, Q1, Median, Q3, Maximum
Box-and-Whisder plot – represents the five-number summary graphically.

29. Box-and-Whisker Plot Construction

30. Critical Thinking
Box-and-whisker plots display the spread of data about the median.
If the median is centered and the whiskers are about the same length, then the data distribution is symmetric around the median.
Fences – may be placed on either side of the box. Values lie beyond the fences are outliers. (See problem 10)

31. Critical Thinking
Which of the following box-and-whiskers plots suggests a symmetric data distribution?
(a) (b) (c) (d)

32. Critical Thinking
Which of the following box-and-whiskers plots suggests a symmetric data distribution?
(a) (b) (c) (d)