2.1 statics notes 2.1, Lecture notes of Mathematics

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Chapter

Modeling Distributions of

Quantitative Data

Section 2.

Describing Location

in

a Distribution

Starnes/Tabor, The Practice of

By the end of this section, you should be able to: LEARNING TARGETS Describing Location in a Distribution FIND and INTERPRET the percentile of an individual value within a distribution of data. ESTIMATE percentiles and individual values using a cumulative relative frequency graph. FIND and INTERPRET the standardized score ( z-score) of an individual value within a distribution of data. DESCRIBE the effect of adding, subtracting, multiplying by, or dividing by a constant on the shape, center, and variability of a distribution of data.

Starnes/Tabor, The Practice of

Statistics

One way to describe the location of a value in a distribution is to tell what percent of observations are less than it. Measuring Location: Percentiles 6 | 7 7 | 2334 7 | 577789 9 8 | 001233 34 Jenny’s score was greater or equal to 22 of the 25 test scores. Because 22 of the 25, or 88%, of the scores are equal to or less than hers, Jenny’s score is at the 88

th

percentile in the distribution of test scores for the class. Jenny earned a score of 86 on her test. How did she perform relative to the rest of the class? The pth percentile of a distribution is the value with p% of observations less than or equal to it.

Starnes/Tabor, The Practice of

Cumulative Relative Frequency Graphs Age of First 44 Presidents When They Were Inaugurated Age Frequency 40- 44

A cumulative relative frequency graph plots a point corresponding to the percentile of a given value in a distribution of quantitative data. Consecutive points are then connected with a line segment to form the graph.

Starnes/Tabor, The Practice of

Cumulative Relative Frequency Graphs Age of First 44 Presidents When They Were Inaugurated Age Frequency Relative frequency Cumulative frequency 40- 44

A cumulative relative frequency graph plots a point corresponding to the percentile of a given value in a distribution of quantitative data. Consecutive points are then connected with a line segment to form the graph.

Starnes/Tabor, The Practice of

Cumulative Relative Frequency Graphs Age of First 44 Presidents When They Were Inaugurated Age Frequency Relative frequency Cumulative frequency Cumulative relative frequency 40- 44

40 45 50 55 60 65 70 0 20 40 60 80 100

Age at inauguration

Cumulative relative

frequency (%)

A cumulative relative frequency graph plots a point corresponding to the percentile of a given value in a distribution of quantitative data. Consecutive points are then connected with a line segment to form the graph.

Starnes/Tabor, The Practice of

Measuring Position: z-Scores Jenny earned a score of 86 on her test. The class mean is 80 and the standard deviation is 6.07. Find and interpret her standardized score. Jenny’s test score is 0.99 standard deviations greater than the class mean of 80.

Starnes/Tabor, The Practice of

Measuring Position: z-Scores Jenny earned a score of 86 on her test. The class mean is 80 and the standard deviation is 6.07. Find and interpret her standardized score. Jenny’s test score is 0.99 standard deviations greater than the class mean of 80. CAUTION : Do not interpret the z-score as a distance “away” from the mean. Always indicate if the value is greater than (above) or less than (below) the mean.

Starnes/Tabor, The Practice of

Transforming Data Soon after the metric system was introduced in Australia, a group of students was asked to guess the width of their classroom to the nearest meter. The actual width of the room was 13 meters. We can examine the distribution of students’ errors by defining a new variable as follows: error = guess – 13.

Starnes/Tabor, The Practice of

Transforming Data Soon after the metric system was introduced in Australia, a group of students was asked to guess the width of their classroom to the nearest meter. The actual width of the room was 13 meters. We can examine the distribution of students’ errors by defining a new variable as follows: error = guess – 13. a) What shape does the distribution of error have?

Starnes/Tabor, The Practice of

Transforming Data Soon after the metric system was introduced in Australia, a group of students was asked to guess the width of their classroom to the nearest meter. The actual width of the room was 13 meters. We can examine the distribution of students’ errors by defining a new variable as follows: error = guess – 13. a) What shape does the distribution of error have? b) Find the mean and the median of the distribution of error. a) The same shape as the original distribution of guesses: skewed to the right with two distinct peaks.

Starnes/Tabor, The Practice of

Transforming Data Soon after the metric system was introduced in Australia, a group of students was asked to guess the width of their classroom to the nearest meter. The actual width of the room was 13 meters. We can examine the distribution of students’ errors by defining a new variable as follows: error = guess – 13. a) What shape does the distribution of error have? b) Find the mean and the median of the distribution of error. a) The same shape as the original distribution of guesses: skewed to the right with two distinct peaks. b) Mean: 16.02 – 13 = 3. meters; Median: 15 – 13 = 2 meters.

Starnes/Tabor, The Practice of

Statistics

Transforming Data Soon after the metric system was introduced in Australia, a group of students was asked to guess the width of their classroom to the nearest meter. The actual width of the room was 13 meters. We can examine the distribution of students’ errors by defining a new variable as follows: error = guess – 13. a) What shape does the distribution of error have? b) Find the mean and the median of the distribution of error. c) Find the standard deviation and interquartile range ( IQR) of the distribution of error. a) The same shape as the original distribution of guesses: skewed to the right with two distinct peaks. b) Mean: 16.02 – 13 = 3. meters; Median: 15 – 13 = 2 meters. c) Standard deviation = 7. meters; IQR = 6 meters

Starnes/Tabor, The Practice of

Transforming Data