# Sullivan 2nd ed Chapter 3 by dffhrtcv3

VIEWS: 0 PAGES: 14

• pg 1
```									                                   Chapter 3
Section 4
Measures of
Position

Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 3 Section 4 – Slide 1 of 23
Chapter 3 – Section 4

● Mean / median describe the “center” of the data
● Variance / standard deviation describe the
● This section discusses more precise ways to
describe the relative position of a data value
within the entire set of data

Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 3 Section 4 – Slide 2 of 23
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

Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 3 Section 4 – Slide 3 of 23
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

Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 3 Section 4 – Slide 4 of 23
Chapter 3 – Section 4

● The population z-score is calculated using the
population mean and population standard
deviation
x
z


● The sample z-score is calculated using the
sample mean and sample standard deviation
xx
z
s

Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 3 Section 4 – Slide 5 of 23
Chapter 3 – Section 4

● z-scores can be used to compare the relative
positions of data values in different samples
where the mean grade was 74 and the standard
deviation was 12
where the mean grade was 65 and the standard
deviation was 10
where the mean grade was 88 and the standard
deviation was 6

Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 3 Section 4 – Slide 6 of 23
Chapter 3 – Section 4

● Statistics
 z-score of (82 – 74) / 12 = .67
● Biology
 z-score of (72 – 65) / 10 = .70
● Kayaking
 z-score of (91 – 88) / 6 = .50
● Biology was the highest relative grade

Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 3 Section 4 – Slide 7 of 23
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

Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 3 Section 4 – Slide 8 of 23
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

Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 3 Section 4 – Slide 9 of 23
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

Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 3 Section 4 – Slide 10 of 23
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

Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 3 Section 4 – Slide 11 of 23
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
● The fences used to identify outliers are
 Lower fence = LF = Q1 – 1.5  IQR
 Upper fence = UF = Q3 + 1.5  IQR
● Values less than the lower fence or more than
the upper fence could be considered outliers

Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 3 Section 4 – Slide 12 of 23
Chapter 3 – Section 4

● 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 = (27 + 31) / 2 = 29
   IQR = 29 – 5.5 = 23.5
   UF = Q3 + 1.5  IQR = 29 + 1.5  23.5 = 64
● Using the fence rule, the value 54 is not an
outlier

Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 3 Section 4 – Slide 13 of 23
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

Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 3 Section 4 – Slide 14 of 23

```
To top