1. Analyzing One-Variable Data
Lesson 1.9
Describing Location in a Distribution
Statistics and Probability with Applications, 3rd Edition
Starnes, Tabor
Bedford Freeman Worth Publishers

2. Describing Location in a Distribution
Learning Targets
After this lesson, you should be able to:
Find and interpret a percentile in a distribution of quantitative data.
Estimate percentiles and individual values using a cumulative relative frequency graph.
Find and interpret a standardized score (z-score) in a distribution of quantitative data.

3. Describing Location in a Distribution
Here are the scores of all 25 students in Mr. Pryor’s statistics class on their first test: The bold score is Jenny’s 86. How did she perform on this test relative to her classmates?

4. Describing Location in a Distribution
One way to describe Jenny’s location in the distribution of test scores is to calculate her percentile.
Percentile
An individual’s percentile is the percent of values in a distribution that are less than the individual’s data value.
Because 21 of the 25 observations (84%) are below her score, Jenny is at the 84th percentile in the class’s test score distribution.
Be careful with your language when describing percentiles. Percentiles are specific locations in a distribution, so an observation isn’t “in” the 84th percentile. Rather, it is “at” the 84th percentile.

5. Describing Location in a Distribution
There are some interesting graphs that can be made with percentiles. One of the most common starts with a frequency table for a quantitative variable and expands it to include cumulative frequency and cumulative relative frequency.
Cumulative Relative Frequency Graph
A cumulative relative frequency graph plots a point corresponding to the cumulative relative frequency in each interval at the smallest value of the next interval, starting with a point at a height of 0% at the smallest value of the first interval. Consecutive points are then connected with a line segment to form the graph.

6. Describing Location in a Distribution
Cumulative Relative Frequency Graphs
Describing Location in a Distribution
Age of First 44 Presidents When They Were Inaugurated
Age
Frequency
Relative frequency
Cumulative frequency
Cumulative relative frequency
40-44 2
2/44 = 4.5%
2/44 =
4.5%
45-49 7
7/44 = 15.9% 9
9/44 = 20.5%
50-54 13
13/44 = 29.5% 22
22/44 = 50.0%
55-59 12
12/44 = 34% 34
34/44 = 77.3%
60-64 41
41/44 = 93.2%
65-69 3
3/44 = 6.8% 44
44/44 = 100%

7. Describing Location in a Distribution
Age
Frequency
Relative frequency
Cumulative
frequency
Cumulative relative
40–44 2
2/44 = 0.045, or 4.5%
45–49 7
7/44 = 0.159, or 15.9% 9
9/44 = 0.205, or 20.5%
50–54 13
13/44 = 0.295, or 29.5% 22
22/44 = 0.500, or 50.0%
55–59 12
12/44 = 0.273, or 27.3% 34
34/44 = 0.773, or 77.3%
60–64 41
41/44 = 0.932, or 93.2%
65–69 3
3/44 = 0.068, or 6.8% 44
44/44 = 1.000, or 100%
A cumulative relative frequency graph can be used to describe the position of an individual within a distribution or to locate a specified percentile of the distribution.

8. Describing Location in a Distribution
A cumulative relative frequency graph (or ogive) displays the cumulative relative frequency of each class of a frequency distribution.
Was Barack Obama, who was inaugurated at age 47, unusually young?
Estimate and interpret the 65th percentile of the distribution
Describing Location in a Distribution 65 11 58 47

9. Describing Location in a Distribution
A percentile is one way to describe the location of an individual in a distribution of quantitative data. Another way is to give the standardized score (z-score) for the observed value.
Standardized Score (z-score)
The standardized score (z-score) for an individual value in a distribution tells us how many standard deviations from the mean the value falls, and in what direction. To find the standardized score (z-score), compute

10. The Standard Deviation as a Ruler
The trick in comparing very different-looking values is to use standard deviations as our rulers.
The standard deviation tells us how the whole collection of values varies, so it’s a natural ruler for comparing an individual to a group.
As the most common measure of variation, the standard deviation plays a crucial role in how we look at data.

11. Standardizing with z-scores (cont.)
The standardized score (z-score) for an individual value in a distribution tells us how many standard deviations from the mean the value falls, and in what direction. To find the standardized score (z-score), compute
Standardized values have no units.
z-scores measure the distance of each data value from the mean in standard deviations.
A negative z-score tells us that the data value is below the mean, while a positive z-score tells us that the data value is above the mean.

12. Benefits of Standardizing
Standardized values have been converted from their original units to the standard statistical unit of standard deviations from the mean.
Thus, we can compare values that are measured on different scales, with different units, or from different populations.
A z-score gives us an indication of how unusual a value is because it tells us how far it is from the mean.
Remember that a negative z-score tells us that the data value is below the mean, while a positive z-score tells us that the data value is above the mean.
The larger a z-score is (negative or positive), the more unusual it is.

13. Describing Location in a Distribution
Values larger than the mean have positive z-scores. Values smaller than the mean have negative z-scores. Let’s return to the data from Mr. Pryor’s first statistics test. Where does Jenny’s 86 fall within the distribution?
Her standardized score (z-score) is
That is, Jenny’s test score is 0.99 standard deviations above the mean score of the class.

14. Median household income
LESSON APP 1.9
Which states are rich?
The following cumulative relative frequency graph and the numerical summaries describe the distribution of median household incomes in the 50 states in a recent year.
At what percentile is North Dakota, with a median household income of $55,766?
Estimate and interpret the first quartile Q1 of the distribution.
Find and interpret the standardized score (z-score) for New Jersey, with a median household income of $66,692.
Median household income n 50
Mean
$51,742.44 SD
$8,

15. Describing Location in a Distribution
Learning Targets
After this lesson, you should be able to:
Find and interpret a percentile in a distribution of quantitative data.
Estimate percentiles and individual values using a cumulative relative frequency graph.
Find and interpret a standardized score (z-score) in a distribution of quantitative data.