Section 2.5 Measures of Position
Quartiles, Percentiles, Standard Score (z score)
Quartiles, deciles, and percentiles divide the data set into equal parts.
The data must be arranged in order to find these measures of position.
Fractiles
Quartiles
Summary
Divide a data set into four equal parts.
Symbols
Q1 , Q2 , Q3
Deciles
Divide a data set into ten equal parts.
D1 , D2 , D3 ... D9
Percentiles Divide a data set into one hundred equal parts.
P1 , P2 , P3 ... P99
QUARTILES:
Q2 is equal to the median of the data set.
Q1 is the median of the values that are below Q2 .
Q3 is the median of the values that are above Q2 .
Find the quartiles for the data set below.
2, 14, 17, 18, 19, 20, 21, 21, 21, 22, 23, 25
Quartiles are used to draw box-and-whisker plots.
The interquartile range is defined as IQR
of the data.
Q3
Q1 , this is the range of the middle 50%
The interquartile range can be used to identify outliers in a data set.
If a data value is less than Q1 1.5 IQR or greater than Q3 1.5 IQR , it is considered an
outlier. Check the above data set for outliers.
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PERCENTILES:
If a test score of 85 corresponds to P95 , that test score is higher than or as high as
95% of the scores in the data set.
If a test score of 85 corresponds to P75 ( equal to Q3 ), that test score is higher than or
as high as 3/4 of the scores in the data set.
To find the percentile for a given data value, x:
percentile
# of values less than x
100
total # of values
Round to the nearest whole number.
Find the percentile for 19 in the set above.
Use the relative ogive graph to estimate
a)
the percentile that corresponds to a score of 1200.
b)
the score that corresponds to P90
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A z score for a data value is found by this formula:
z
For population data
X
z
value mean
st . deviation
For sample data
X X
z
s
The z score represents how many standard deviations a value is from the mean of the
data set. If the z score for a data value is positive, that value is greater than the
mean of the data set. If the z score for a data value is zero, that value is equal to the
mean of the data set. If the z score for a data value is negative, that value is less than
the mean of the data set. Rounding rule: Round z scores to two decimal places.
Practice Finding z scores
For a Statistics test
X = 63.0 points, s = 7.0 points
For a Biology test
X = 23.0 points, s = 3.9 points
a) Find the z score for each test score.
b) Determine on which test the student had a better score.
1)
A student gets 73 on the statistics test and 26 on the biology test.
2)
A student gets 60 on the statistics test and 22 on the biology test.
The Empirical (Normal) Rule
When a distribution is bell shaped (normal):
Approximately 68% of the data fall within 1 standard deviation of the mean.
Approximately 95% of the data fall within 2 standard deviations of the mean.
Approximately 99.7% of the data fall within 3 standard deviations of the mean.
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