Download Section 3.3 Relative Position Percentile

Section 3.3 Relative Position
In this section we look at different ways to measure the relative position of things.
What if you made a 62 on a test?
On the surface it might look bad but what if that was the highest in the class or if that score
was better than 80% of the class.
This is what we mean by relative position.
z-Score
A z-score gives us and estimate as to how many standard deviations a particular data value
lies from the mean. We define it as
X  the data value in question
X  the sample mean
s  the sample standard deviation
z  X −s X
Examples 3-29 and 3-30 in the text illustrate this.
If the z-score is 0 then your data value is the mean
If the z-score  0 (positive) then your data value is above the mean
If the z-score  0 (negative) then your data value is the below the mean.
Percentile
This the measurement that you often see at the doctor office for height as seen on page 153.
You also often see these items in standardized tests.
Given a particular test score it tells you what percentage of people you did better than.
Convert Data Value to Percentile
1. Arrange the data in ascending order
2. Count how many items are below your value. If for example your score is and 85 and
there a multiple 85’s then count how many are under the first 85.
For example
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76, 77, 80, 83, 85, 85, 85, 90, 96, 97
There are 4 items below 85
Use the formula on page 155
Percentile 
number of items below your data  0. 5
∗ 100%
total number of values
So in our data example
Percentile  4  0. 5 ∗ 100%  45 Percentile
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Percentile to What Data Value
In this case someone hands us a percentile such 30 and then asks which data value does this
correspond to.
1. Arrange the data in ascending order.
2. Calculate
total number of values * percentile
c
100
In our example using the 30th percentile
c  10 ∗ 30  3
100
The value of c is a position in the data and it will either be a whole number as in our
example or it will have a decimal part.
If c has a decimal part, suppose c  3.45 then you round it up to the closest whole number
which would be 4 which is the fourth position.
If c is a whole number like in our example where we got c  3 then you go to the numbers
in the third position (c  3) and the fourth position (c  1  4) and average those 2 values.
76, 77, 80, 83, 85, 85, 85, 90, 96, 97
For our example
position 1  76
position 2  77
position 3  80
position 4  83.
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80  83  81. 5
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and this is the 30th percentile.
If c had been  4.5 then we would round to 5 and get the data value at position 5 which is
85
Quartiles and Outliers
We will talk about these topics in the next section in the context of Box Plots.
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