1. Measures of Position & Exploratory Data Analysis
Organizing Data
Measures of Position &
Exploratory Data Analysis
Essentials
Measures of Position
Five-Number-Summary
Example
Box and Whiskers Plots
Pearson’s Index of Skewness
Z-scores
Coefficient of Variation

2. Essentials: Measures of Position (Better understanding distribution shapes.)
Know the types of measures used to look at specific positions within a data distribution.
Be able to calculate the inter-quartile range, three quartiles, Pearson’s Index of Skewness, z-score, Coefficient of Variation.
Be familiar with symmetry vs. skewness and distribution shapes.
Be able to build both traditional and modified box plots (aka: box-and-whiskers plot).

3. Measures of Position
Measures of position are points within the data that are used to describe characteristics of the data. Percentiles, Deciles, Quartiles, Minimum and Maximum are among these points.
We will focus on 5 specific values...

4. The Five-Number-Summary
Five numbers are frequently used to indicate positions within a data set and much more...
These points are:
Minimum - Q1 - Median - Q3 - Maximum

5. Components of the Five-Number Summary
The Minimum and Maximum values represent the extremes in the data set.
Obtaining these points is a simple matter of looking for them.
Q1, Median (Q2), and Q3 represent points within the data. They are the 25th, 50th, and 75th percentiles, respectively.
Formulas are used to locate these positions.

6. The Quartiles
The Quartiles are obtained using the following three formulas.
Q1(25th percentile) = (n+1)/4
Median (Q2, 50th percentile) = (n+1)/2
Q3 (75th percentile) = [3(n+1)]/4
(Where n = number of observations. Note that these formulas identify POSITIONS within the data, not values; the data must be placed in numeric order)

7. The Interquartile Range
The Interquartile Range (IQR) describes the middle 50% of the data. It is obtained by the formula:
IQR = Q3 - Q1
The Interquartile Range is used as a measure of variation around the Median of a set of data.

8. Anatomy of Measures of Position
The Minimum value, First Quartile (Q1), Median (Q2), Third Quartile (Q3) and Maximum value comprise what is referred to as the Five Number Summary. Each component of this summary represents a measure of position within a set of data. Together, these five values are referred to as the Five Number Summary.
Quartiles divide a data set that has been
ordered from smallest to largest into four
sections, each containing 25% of the data values.
A term more familiar to you might be percentile.
For example, the 25th percentile,
50th percentile, or 75th percentile.
Anatomy of Measures of Position
Recall that the median in this data set is found
using the formula (n+1)/2 to obtain the POSITION of the median
(here the POSITION is (20 + 1)/2 = 10.5). Determining the value half way
between the 10th and 11th values yields a Median of 558: [( )/2 = 558].
Another name for the median, is the second quartile (Q2) . It is the value
in the data set such that 50% of the values are lower than it,
and 50% of the values are higher than it.
The third quartile (Q3) is the value
such that 75% of the values are lower
than it, and 25% of the values are higher than it.
To find this value apply the formula [3(n+1)/4] to obtain
the POSITION of the third quartile. Here the formula yields
POSITION 15.75: [(3(20+1))/4 = 15.75]. Determine the value ¾
of the way between the 15th value (638) and the 16th value (664)
by determining the difference between these two
numbers ( = 26); multiplying the
difference by .75 (26 * .75 = 19.5); and adding
this value to the smaller number
( = = Q3)
The first quartile (Q1) is the value
such that 25% of the values are lower
than it, and 75% of the values are higher than it.
To find this value apply the formula (n+1)/4 to obtain
the POSITION of the first quartile. Here the formula yields
POSITION 5.25: [(20+1)/4 = 5.25]. Determine the value ¼
of the way between the 5th value (483) and the 6th value (514)
by determining the difference between these two
numbers (514 – 483 = 31); multiplying the
difference by .25 (31 * .25 = 7.75); and adding
this value to the smaller number
( = = Q1)
The Interquartile Range is the
difference between the third quartile, and the first quartile. Here, the
interquartile range is =

9. Exploring and Comparing Data
Exploratory Data Analysis (EDA): EDA is the process of using statistical tools (graphs, measures of center, variation, and position) to investigate data sets in order to understand their important characteristics.

10. Box-and-Whiskers Plot
a.k.a. Box plot
A Box plot is used to display the five-number-summary. One can also examine the shape of the distribution with a Box plot.
Data Presentation:
Box plot: Traditional vs. Modified Box plot
Display of Outliers and Adjacent Values

11. Anatomy of a Traditional Box plot
DUDLEY’S DOUGHNUTS
(flour in pounds, used on 20 consecutive days during the
month of Dec. 1999)
Title
This is Q3, the
third quartile. Here,
Q3 is
This is the Upper
Whisker. It is the
maximum value in the
data set. Here, the
maximum value is 707.
This is the
Median (Q2). Here, the
median is 558.
This is the
Lower Whisker. It is the
minimum value in the data
set. Here, the minimum
value is 440.
This is Q1, the first
quartile. Here, Q1 is
Flour (in lbs.) Used
440, 481, 482, 483, 483, 514, 514, 554, 554, 554,
562, 612, 623, 631, 638, 664, 671, 677, 690, 707
Historical Note
Who invented this useful tool for quick data
analysis?
John Tukey
Statistician

12. Outliers, Limits, and Adjacent Points
An Outlier is a data point found at one of the extremes of the data, and is well outside the general pattern of the data.
Upper and Lower Limits: Used as a tool to identify observations that may be outliers.
Lower Limit = Q (IQR)
Upper Limit = Q (IQR)
Adjacent Points: The last data value that occurs before (or at) the Upper or Lower Limit. In modified Box plots the whisker would stop at this data value rather than being drawn out to 1.5(IQR).

13. Mean Price of a Movie Ticket for a Sample of 12 U.S. Cities
Example of a Modified Box Plot

14. Grouped Box plots
A grouped box plot is a good way to visualize differences/similarities between groups.
SPSS Data Set: cars.sav

15. Pearson’s Index of Skewness
Skewness can be measured using Pearson’s Index of Skewness.
Example: Given a set of date whose statistics include: mean = 40, median = 41, S.D = 4, determine if this distribution id skewed.
I = (3( ))/4 = -.75
Given that the value is within the range from -1 to 1, inclusive, this distribution would not be considered to be significantly skewed.

16. Pearson’s Index of Skewness
When symmetric, I = 0.
Values usually range from –3 to +3.
A distribution is considered symmetric if the index value is between -1 and +1
If the index value (I) is less than -1 the data are negatively (left) skewed.
If the index value is greater than +1 The data are positively (right) skewed.

17. Pearson’s Index of Skewness: Example
Use Pearson’s Index of Skewness to determine if the distribution of 406 automobile
weights is approximately normally distributed or does it display a degree of skewness?
Mean Wt: lb.
Median Wt: lb.
Standard Deviation: lb.

18. Measures of Position
Standard Scores: a standard score, or z-score is the number of standard deviations that a given value x is above or below the mean. To find a z-score
For populations: For samples:
(Always round z to two decimal places.)

19. Standard Scores (z-score)
Recall, a z-score is a measure of a value’s distance away from a distribution’s mean as measured in standard deviations.
Example: Ozzie just took two tests. Given his scores, the mean for the tests, and the standard deviations, on which test did Ozzie perform better relative to the other students?
Calculus Exam: Grade = 65, class mean = 50, S.D. = 10
z = ( )/10 = 1.5 (or 1.5 standard deviations above the mean)
History exam: Grade = 30, class mean = 25, S.D. = 5
z = ( )/5 = 1 (or 1 standard deviation above the mean)
Since the z-score for the calculus exam is larger, Ozzie’s relative position is higher in the calculus class than it is in the history class.

20. Coefficient of Variation
Allows us to compare standard deviations. The result is expressed as a percentage.
Example:Trinity’s test statistics included: Anthropology test - mean of 50 and S.D. of 10; Music test - mean of 40 and S.D. of 5. Which test showed greater variation in test scores?
Anthropology: (10/50)*100 = 20%
Music: (5/40)*100 = 12.5%
Thus, there was greater variation in test scores for the Anthropology test.
Use example 3-26 on pg. 120 of text.

21. Coefficient of Variation: Example
The heights and weights and ages of the starting members of the 2008 World Champion Boston Celtics are noted below. Determine the coefficient of determination for these variables to determine which has the greatest variation.
Ray Allen: 77 in., 205 lb., 33 yrs.
Rajon Rondo: 73 in., 171 lb., 21 yrs.
Paul Pierce: 79 in., 235 lb., 30 yrs.
Kevin Garnett: 83 in., 253 lb., 32 yrs.
Kendrick Perkins: 82 in., 280 lb., 23 yrs.