1. Chapter 3 Section 4
Measures of
Position

2. Chapter 3 – Section 4 Mean / median describe the “center” of the data
Variance / standard deviation describe the “spread” of the data
This section discusses more precise ways to describe the relative position of a data value within the entire set of data

3. Chapter 3 – Section 4
The standard deviation is a measure of dispersion that uses the same dimensions as the data (remember the empirical rule)
The distance of a data value from the mean, calculated as the number of standard deviations, would be a useful measurement
This distance is called the z-score
The standard deviation is a measure of dispersion that uses the same dimensions as the data (remember the empirical rule)
The standard deviation is a measure of dispersion that uses the same dimensions as the data (remember the empirical rule)
The distance of a data value from the mean, calculated as the number of standard deviations, would be a useful measurement

4. Chapter 3 – Section 4
If the mean was 20 and the standard deviation was 6
The value 26 would have a z-score of 1.0 (1.0 standard deviation higher than the mean)
The value 14 would have a z-score of –1.0 (1.0 standard deviation lower than the mean)
The value 17 would have a z-score of –0.5 (0.5 standard deviations lower than the mean)
The value 20 would have a z-score of 0.0
If the mean was 20 and the standard deviation was 6
The value 26 would have a z-score of 1.0 (1.0 standard deviation higher than the mean)
The value 14 would have a z-score of –1.0 (1.0 standard deviation lower than the mean)
The value 17 would have a z-score of –0.5 (0.5 standard deviations lower than the mean)
If the mean was 20 and the standard deviation was 6
The value 26 would have a z-score of 1.0 (1.0 standard deviation higher than the mean)
If the mean was 20 and the standard deviation was 6
If the mean was 20 and the standard deviation was 6
The value 26 would have a z-score of 1.0 (1.0 standard deviation higher than the mean)
The value 14 would have a z-score of –1.0 (1.0 standard deviation lower than the mean)

5. Chapter 3 – Section 4
The population z-score is calculated using the population mean and population standard deviation
The sample z-score is calculated using the sample mean and sample standard deviation
The population z-score is calculated using the population mean and population standard deviation

6. Chapter 3 – Section 4
z-scores can be used to compare the relative positions of data values in different samples
Pat received a grade of 82 on her statistics exam where the mean grade was 74 and the standard deviation was 12
Pat received a grade of 72 on her biology exam where the mean grade was 65 and the standard deviation was 10
Pat received a grade of 91 on her kayaking exam where the mean grade was 88 and the standard deviation was 6
z-scores can be used to compare the relative positions of data values in different samples
Pat received a grade of 82 on her statistics exam where the mean grade was 74 and the standard deviation was 12
Pat received a grade of 72 on her biology exam where the mean grade was 65 and the standard deviation was 10
z-scores can be used to compare the relative positions of data values in different samples
Pat received a grade of 82 on her statistics exam where the mean grade was 74 and the standard deviation was 12
z-scores can be used to compare the relative positions of data values in different samples

7. Chapter 3 – Section 4 Statistics Biology Kayaking
Grade of 82
z-score of (82 – 74) / 12 = .67
Biology
Grade of 72
z-score of (72 – 65) / 10 = .70
Kayaking
Grade of 81
z-score of (91 – 88) / 6 = .50
Biology was the highest relative grade
Statistics
Grade of 82
z-score of (82 – 74) / 12 = .67
Biology
Grade of 72
z-score of (72 – 65) / 10 = .70
Kayaking
Grade of 81
z-score of (91 – 88) / 6 = .50
Statistics
Grade of 82
z-score of (82 – 74) / 12 = .67
Biology
Grade of 72
z-score of (72 – 65) / 10 = .70
Statistics
Grade of 82
z-score of (82 – 74) / 12 = .67

8. Chapter 3 – Section 4
The quartiles are the 25th, 50th, and 75th percentiles
Q1 = 25th percentile
Q2 = 50th percentile = median
Q3 = 75th percentile
Quartiles are the most commonly used percentiles
The 50th percentile and the second quartile Q2 are both other ways of defining the median

9. Chapter 3 – Section 4
Quartiles divide the data set into four equal parts
The top quarter are the values between Q3 and the maximum
The bottom quarter are the values between the minimum and Q1
Quartiles divide the data set into four equal parts
The top quarter are the values between Q3 and the maximum
Quartiles divide the data set into four equal parts
Quartiles divide the data set into four equal parts
Quartiles divide the data set into four equal parts

10. Chapter 3 – Section 4
Quartiles divide the data set into four equal parts
The interquartile range (IQR) is the difference between the third and first quartiles
IQR = Q3 – Q1
The IQR is a resistant measurement of dispersion

11. Chapter 3 – Section 4
Extreme observations in the data are referred to as outliers
Outliers should be investigated
Outliers could be
Chance occurrences
Measurement errors
Data entry errors
Sampling errors
Outliers are not necessarily invalid data
Extreme observations in the data are referred to as outliers
Outliers should be investigated
Extreme observations in the data are referred to as outliers
Outliers should be investigated
Outliers could be
Chance occurrences
Measurement errors
Data entry errors
Sampling errors

12. Chapter 3 – Section 4 One way to check for outliers uses the quartiles
Outliers can be detected as values that are significantly too high or too low, based on the known spread
The fences used to identify outliers are
Lower fence = LF = Q1 – 1.5  IQR
Upper fence = UF = Q  IQR
Values less than the lower fence or more than the upper fence could be considered outliers
One way to check for outliers uses the quartiles
Outliers can be detected as values that are significantly too high or too low, based on the known spread
One way to check for outliers uses the quartiles
Outliers can be detected as values that are significantly too high or too low, based on the known spread
The fences used to identify outliers are
Lower fence = LF = Q1 – 1.5  IQR
Upper fence = UF = Q  IQR

13. Chapter 3 – Section 4 Is the value 54 an outlier? Calculations
1, 3, 4, 7, 8, 15, 16, 19, 23, 24, 27, 31, 33, 54
Calculations
Q1 = (4 + 7) / 2 = 5.5
Q3 = ( ) / 2 = 29
IQR = 29 – 5.5 = 23.5
UF = Q  IQR =  23.5 = 64
Using the fence rule, the value 54 is not an outlier
Is the value 54 an outlier?
1, 3, 4, 7, 8, 15, 16, 19, 23, 24, 27, 31, 33, 54
Calculations
Q1 = (4 + 7) / 2 = 5.5
Q3 = ( ) / 2 = 29
IQR = 29 – 5.5 = 23.5
UF = Q  IQR =  23.5 = 64
Is the value 54 an outlier?
1, 3, 4, 7, 8, 15, 16, 19, 23, 24, 27, 31, 33, 54
Calculations
Q1 = (4 + 7) / 2 = 5.5
Q3 = ( ) / 2 = 29
Is the value 54 an outlier?
1, 3, 4, 7, 8, 15, 16, 19, 23, 24, 27, 31, 33, 54

14. Summary: Chapter 3 – Section 4
z-scores
Measures the distance from the mean in units of standard deviations
Can compare relative positions in different samples
Percentiles and quartiles
Divides the data so that a certain percent is lower and a certain percent is higher
Outliers
Extreme values of the variable
Can be identified using the upper and lower fences