1. Analyzing One-Variable Data1
Analyzing One-Variable Data
Lesson 1.9
Describing Location in a Distribution

2. Describing Location in a Distribution
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
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.

7. 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

8. 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.

9. 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.

10. Describing Location in a Distribution
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.