Download Sullivan 2nd ed Chapter 3

Chapter 3
Section 4
Measures of
Position
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 3 Section 4 – Slide 1 of 23
Chapter 3 – Section 4
● Learning objectives
1

Determine and interpret z-scores
2 Determine and interpret percentiles
3
 Determine and interpret quartiles
4
 Check a set of data for outliers
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 3 Section 4 – Slide 2 of 23
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
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 3 Section 4 – Slide 3 of 23
Chapter 3 – Section 4
● Learning objectives
1

Determine and interpret z-scores
2 Determine and interpret percentiles
3
 Determine and interpret quartiles
4
 Check a set of data for outliers
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 3 Section 4 – Slide 4 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 – Fundamentals of Statistics – 2nd Edition – Chapter 3 Section 4 – Slide 5 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 – Fundamentals of Statistics – 2nd Edition – Chapter 3 Section 4 – Slide 6 of 23
Chapter 3 – Section 4
● The population z-score is calculated using the
population mean and population standard
deviation
z
x

● The sample z-score is calculated using the
sample mean and sample standard deviation
xx
z
s
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 3 Section 4 – Slide 7 of 23
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
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 3 Section 4 – Slide 8 of 23
Chapter 3 – Section 4
● 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
● Biology was the highest relative grade
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 3 Section 4 – Slide 9 of 23
Chapter 3 – Section 4
● Learning objectives
1

Determine and interpret z-scores
2 Determine and interpret percentiles
3
 Determine and interpret quartiles
4
 Check a set of data for outliers
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 3 Section 4 – Slide 10 of 23
Chapter 3 – Section 4
● The median divides the lower 50% of the data
from the upper 50%
● The median is the 50th percentile
● If a number divides the lower 34% of the data
from the upper 66%, that number is the 34th
percentile
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 3 Section 4 – Slide 11 of 23
Chapter 3 – Section 4
● The computation is similar to the one for the
median
● Calculation
 Arrange the data in ascending order
 Compute the index i using the formula
k 

i 
 n  1
 100 
● If i is an integer, take the ith data value
● If i is not an integer, take the mean of the two
values on either side of i
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 3 Section 4 – Slide 12 of 23
Chapter 3 – Section 4
● Compute the 60th percentile of
1, 3, 4, 7, 8, 15, 16, 19, 23, 24, 27, 31, 33, 34
● Calculations
 There are 14 numbers (n = 14)
 The 60th percentile (k = 60)
 The index
k 
 60  14  1  9
i  

n

1





 100 
 100 
● Take the 9th value, or P60 = 23, as the 60th
percentile
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 3 Section 4 – Slide 13 of 23
Chapter 3 – Section 4
● Compute the 28th percentile of
1, 3, 4, 7, 8, 15, 16, 19, 23, 24, 27, 31, 33, 54
● Calculations
 There are 14 numbers (n = 14)
 The 28th percentile (k = 28)
 The index
k 
28 


i 
 n  1  
 14  1  4.2
 100 
 100 
● Take the average of the 4th and 5th values, or
P28 = (7 + 8) / 2 = 7.5, as the 28th percentile
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 3 Section 4 – Slide 14 of 23
Chapter 3 – Section 4
● Learning objectives
1

Determine and interpret z-scores
2 Determine and interpret percentiles
3
 Determine and interpret quartiles
4
 Check a set of data for outliers
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 3 Section 4 – Slide 15 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 – Fundamentals of Statistics – 2nd Edition – Chapter 3 Section 4 – Slide 16 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 – Fundamentals of Statistics – 2nd Edition – Chapter 3 Section 4 – Slide 17 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 – Fundamentals of Statistics – 2nd Edition – Chapter 3 Section 4 – Slide 18 of 23
Chapter 3 – Section 4
● Learning objectives
1

Determine and interpret z-scores
2 Determine and interpret percentiles
3
 Determine and interpret quartiles
4
 Check a set of data for outliers
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 3 Section 4 – Slide 19 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 – Fundamentals of Statistics – 2nd Edition – Chapter 3 Section 4 – Slide 20 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
known spread
● 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 – Fundamentals of Statistics – 2nd Edition – Chapter 3 Section 4 – Slide 21 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 probably
not an outlier
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 3 Section 4 – Slide 22 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 – Fundamentals of Statistics – 2nd Edition – Chapter 3 Section 4 – Slide 23 of 23