1. Numerical Descriptive Measures
Chapter 2
Borrowed from

2. Chapter Topics Measures of central tendency Quartiles
Mean, median, mode, midrange
Quartiles
Measure of variation
Range, interquartile range, variance and Standard deviation, coefficient of variation
Shape
Symmetric, skewed (+/-)
Coefficient of correlation

3. Coefficient of Variation
Summary Measures
Summary Measures
Central Tendency
Quartile
Variation
Arithmetic Mean
Median
Mode
Coefficient of Variation
Range
Variance
Geometric Mean
Standard Deviation

4. Measures of Central Tendency
Average (Mean)
Median
Mode

5. Mean (Arithmetic Mean)
Mean (arithmetic mean) of data values
Sample mean
Population mean
Sample Size
Population Size

6. Mean (Arithmetic Mean)
(continued)
The most common measure of central tendency
Affected by extreme values (outliers)
Mean = 5
Mean = 6

7. Median Robust measure of central tendency
Not affected by extreme values
In an Ordered array, median is the “middle” number
If n or N is odd, median is the middle number
If n or N is even, median is the average of the two middle numbers
Median = 5
Median = 5

8. Mode A measure of central tendency Value that occurs most often
Not affected by extreme values
Used for either numerical or categorical data
There may may be no mode
There may be several modes
No Mode
Mode = 9

9. Quartiles Split Ordered Data into 4 Quarters Position of i-th Quartile
and Are Measures of Noncentral Location
= Median, A Measure of Central Tendency
25%
25%
25%
25%
Data in Ordered Array:

10. Coefficient of Variation
Measures of Variation
Variation
Variance
Standard Deviation
Coefficient of Variation
Range
Population
Variance (σ2)
Population Standard deviation (σ)
Inter-quartile Range
Sample
Variance (S2)
Sample Standard deviation (S)

11. Range Measure of variation
Difference between the largest and the smallest observations:
Ignores the way in which data are distributed
Range = = 5
Range = = 5

12. Interquartile Range Measure of variation Also known as midspread
Spread in the middle 50%
Difference between the first and third quartiles
Not affected by extreme values
Data in Ordered Array:

13. Variance Important measure of variation Shows variation about the mean
Sample variance:
Population variance:

14. Standard Deviation Most important measure of variation
Shows variation about the mean
Has the same units as the original data
Sample standard deviation:
Population standard deviation:

15. Comparing Standard Deviations
Data A
Mean = 15.5
S = 3.338
Data B
Mean = 15.5
S = .9258
Data C
Mean = 15.5
S = 4.57

16. Coefficient of Variation
Measures relative variation
Always in percentage (%)
Shows variation relative to mean
Is used to compare two or more sets of data measured in different units

17. Comparing Coefficient of Variation
Stock A:
Average price last year = $50
Standard deviation = $5
Stock B:
Average price last year = $100
Coefficient of variation:

18. Shape of a Distribution
Describes how data is distributed
Measures of shape
Symmetric or Skewed
(-) Left-Skewed
Symmetric
(+) Right-Skewed
Mean < Median < Mode
Mean = Median =Mode
Mode < Median < Mean

19. Coefficient of Correlation
Measures the strength of the linear relationship between two quantitative variables

20. Features of Correlation Coefficient
Unit free
Ranges between –1 and 1
The closer to –1, the stronger the negative linear relationship
The closer to 1, the stronger the positive linear relationship
The closer to 0, the weaker any positive linear relationship

21. Scatter Plots of Data with Various Correlation CoefficientsY X
r = -1
r = -.6
r = 0
r = .6
r = 1

22. Chapter Summary Described measures of central tendency
Mean, median, mode, midrange
Discussed quartile
Described measure of variation
Range, interquartile range, variance and standard deviation, coefficient of variation
Illustrated shape of distribution
Symmetric, Skewed
Discussed correlation coefficient

23. Stem-n-leaf plot
A class took a test. The students' got the following scores.
First let us draw the stem and leaf plot
This makes it easy to see that the mode, the most common score is an 85 since there are three of those scores and all of the other scores have frequencies of either one or tow.
It will also make it easier to rank the scores

24. Stem-n-leaf
This will make it easier to find the median and the quartiles. There are as many scores above the median as below. Since there are ten scores, and ten is an even number, we can divide the scores into two equal groups with no scores left over. To find the median, we divide the scores up into the upper five and the lower five.

25. Box Plot
The five-number summary is an abbreviated way to describe a sample. The five number summary is a list of the following numbers:
Minimum
First (Lower) Quartile,
Median,
Third (Upper) Quartile,
Maximum
The five number summary leads to a graphical representation of a distribution called the boxplot. Boxplots are ideal for comparing two nearly-continuous variables. To draw a boxplot (see the example in the figure below), follow these simple steps:
The ends of the box (hinges) are at the quartiles, so that the length of the box is the .

26. Box plots The median is marked by a line within the box.
The two vertical lines (called whiskers) outside the box extend to the smallest and largest observations within of the quartiles.
Observations that fall outside of are called extreme outliers and are marked, for example, with an open circle. Observations between and are called mild outliers and are distinguished by a different mark, e.g., a closed circle.

27. Example of boxplot
The Density of Nitrogen - A Comparison of Two Samples
Lord Raleigh was one of the earliest scientists to study the density of nitrogen. In his studies, he noticed something peculiar. The density of nitrogen produced from chemical compounds tended to be smaller than the density of nitrogen produced from the air
However, he was working with fairly small samples, and the question is, was he correct in his conjecture?
Lord Raleigh's measurements which first appeared in Proceedings, Royal Society (London, 55, 1894 pp ) are produced below. The units are the mass of nitrogen filling a certain flask under specified pressure and temperature.
Calculate the summary statistics for each set of data.
Construct side-by-side box plots.

28. Chemical
Atmospheric
2.2989
2.3101
2.2994

29. Box Plot
Atmospheric pressure has 9 observations and chemical pressure has 10 values. Use following five points to draw box plots and compare the two variables
Max
Min Q
Median
2.3101 Q