Measures of Position Section 2-6 M A R I O F. T R I O L A Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman

Measure of Position

Measure of Position Z Score (or standard score) the number of standard deviations a score is above or below the mean

Measure of Position z-score Sample x – x z = s

Measure of Position z-score Sample Population x – µ x – x z = z = s s

round to 2 decimal places Measure of Position z-score Sample Population x – µ x – x z = z = s s round to 2 decimal places

Interpreting Z Scores Z FIGURE Unusual Values Ordinary Values Unusual – 3 – 2 – 1 1 2 3 Z

Example

Quartiles, Deciles, Percentiles Measures of Position Quartiles, Deciles, Percentiles

Quartiles

Quartiles Q1, Q2, Q3

divides ranked scores into four equal parts Quartiles Q1, Q2, Q3 divides ranked scores into four equal parts

divides ranked scores into four equal parts Quartiles Q1, Q2, Q3 divides ranked scores into four equal parts 25% 25% 25% 25% Q1 Q2 Q3

divides ranked data into ten equal parts Deciles D1, D2, D3, D4, D5, D6, D7, D8, D9 divides ranked data into ten equal parts

divides ranked data into ten equal parts Deciles D1, D2, D3, D4, D5, D6, D7, D8, D9 divides ranked data into ten equal parts 10% D1 D2 D3 D4 D5 D6 D7 D8 D9

k th Percentiles (for a given number k)

Quartiles, Deciles, Percentiles Fractiles

Quartiles, Deciles, Percentiles Fractiles partitions data into approximately equal parts

Finding the Percentile of a Given Score

Finding the Percentile of a Given Score number of scores less than x Percentile of score x = • 100 total number of scores

Finding the Score Given a Percentile

Finding the Score Given a Percentile k L = • n n number of scores in the data set k percentile being used L locator that gives the position of a score Pk kth percentile k L = • n 100

Finding the Value of the kth Percentile Start Rank the data. (Arrange the data in order of lowest to highest.) Finding the Value of the kth Percentile Compute L = n where n = number of scores k = percentile in question ) ( k 100 The value of the kth percentile is midway between the Lth score and the next higher score in the original set of data. Find Pk by adding the L th score and the next higher score and dividing the total by 2. Is L a whole number ? Yes No Change L by rounding it up to the next larger whole number. The value of Pk is the Lth score, counting from the lowest

Quartiles Q1 = P25 Q2 = P50 Q3 = P75

Quartiles Deciles Q1 = P25 Q2 = P50 Q3 = P75 D1 = P10 D2 = P20 D3 = P30 • D9 = P90

Interquartile Range: Q3 – Q1 Semi-interquartile Range: Midquartile: 10 -90 Percentile Range: P90 - P10 Q3- Q1 2 Q1+ Q3 2

Interquartile Range: Q3 – Q1 Semi-interquartile Range: Midquartile: 10 -90 Percentile Range: P90 - P10 Q3 – Q1 2 Q1+ Q3 2

Interquartile Range: Q3 – Q1 Semi-interquartile Range: Midquartile: 10 -90 Percentile Range: P90 - P10 Q3 – Q1 2 Q1 + Q3 2

Interquartile Range: Q3 – Q1 Semi-interquartile Range: Midquartile: 10–90 Percentile Range: P90 – P10 Q3 – Q1 2 Q1 + Q3 2

Exploratory Data Analysis Section 2-7 M A R I O F. T R I O L A Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman

Exploratory Data Analysis Used to explore data at a preliminary level Few or no assumptions are made about the data Tends to involve relatively simple calculations and graphs

Exploratory Data Analysis Traditional Statistics Used to explore data at a preliminary level Few or no assumptions are made about the data Tends to involve relatively simple calculations and graphs Traditional Statistics Used to confirm final conclusions about data Typically requires some very important assumptions about the data Calculations are often complex, and graphs are often unnecessary

Boxplots Box-and-Whisker Diagram 5 - number summary Minimum first quartile Q1 Median third quartile Q3 Maximum

Boxplots Box-and-Whisker Diagram 60 68.5 78 52 90 50 55 60 65 70 75 80 85 90 Boxplot of Pulse Rates (Beats per minute) of Smokers

Figure 2-14 Boxplots Normal

Figure 2-14 Boxplots Normal Uniform

Figure 2-14 Boxplots Normal Uniform Skewed

Values that are very far away from most of the data Outliers Values that are very far away from most of the data

When comparing two or more boxplots, it is necessary to use the same scale.

When comparing two or more boxplots, it is necessary to use the same scale. 40 50 60 70 80 90 100 PULSE 1 2 (yes) SMOKE (No)

Modified Boxplots Box-and-Whisker Diagram See page 103, Problem 11.