4.4 Measuring Spread Or Variability

Definitions

C^th Percentile

The C^th Percentile of a set of numbers is a value such that at least C% of the numbers fall below it (or equal to it) and at least (100-C)% fall above it (or equal to it).

Lower Quartile

The lower quartile (1^st Quartile or Q1) is the 25^th percentile.

Upper Quartile

The upper quartile (3^rd Quartile or Q3) is the 75^th percentile.

Note 1: The median is the second quartile (Q2) which means that the median is also the 50^th percentile.

Note 2: Also, the lower quartile is the median of the lower half of the data, and the upper quartile is the median of the upper half of the distribution.

Example

Consider the following data set:

The lower quartile is 2 and the upper quartile is 6.

Note: The median is not counted in either half of the distribution.

Definition

5-Number Summary

The 5-number summary of a data set consists of the following descriptive statistics:

1. median
2. lower quartile
3. upper quartile
4. minimum value
5. maximum value.

Boxplot

A boxplot is a graphical representation of the 5-number summary, and it is used as a pictorial description of a data set.

Constructing Boxplots

1. Compute the 5-number summary.

   Data Set: 1 2 2 3 4 5 6 6 6 7 8 9

   Median = 5.5, Q1 = 2.5, Q3 = 6.5, Min = 1, Max = 9

2. Draw a vertical line at the lower and upper quartiles.
   

3. Draw two horizontal lines to complete the box.
   

4. Draw a vertical line at the median.
   

5. Draw "whiskers" to the extremes.
   

Now we will discuss a measure of spread called the standard deviation. The information from the standard deviation is similar to that from a boxplot, but it gives us a measure of spread about the mean of the distribution instead of about the median.

Definitions

Variance

The mean of the squares of the deviations of the observations from their mean is the variance.

Standard Deviation

The positive square root of the variance is the standard deviation.

Finding Variance

1. Complete a variance chart (keep at least 4 decimal places).
   
   

   Variance Table

   Observations		Deviations from		Squared Deviations
   X1		(X1-)		(X1-)2
   X2		(X2-)		(X2-)2
   ...		...		...
   Xn		(Xn-)		(Xn-)2
   		sum = 0.0000		sum of the squared deviations

2. = S².

*** Note the footnote on the bottom of page 222 of the textbook.

Finding Standard Deviation

1. Find the variance = S².

2. Standard Deviation = S = .

Note: Standard deviation makes sense as a measure of spread since it is a kind of average deviation of the observations from their mean.

Homework

(Scott Street's section only)

Pages 223-225

4.46, 4.48, 4.50, 4.51, 4.53, 4.55

(Solutions)

Please direct all questions regarding STAT-110 to your instructor or to the director of STAT-110, Dr. Todd Ogden at ogden@stat.sc.edu.

Mail comments regarding this presentation to W. Scott Street, IV at street@stat.sc.edu.

© 1996 by W. Scott Street, IV