Download MDM4U0 Percentiles Percentiles are similar to quartiles. Quartiles

MDM4U0
Percentiles
Percentiles are similar to quartiles. Quartiles divide a set of data into four groups with equal numbers of
values (just as the median divides data into two equally sized groups.)
Percentiles divide the data into 100 intervals that have equal numbers of values.
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Pk denotes the k percentile. This means that k percent of the data are less than or equal to this value.
Therefore, ___________ percent are greater than or equal to Pk.
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For example: for the 65 percentile, 100 – 65 or 35 percent of the data are greater than or equal to the
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65 percentile.
Example 1
In a neighbourhood of 24 homes, a survey of household incomes (shown in thousands of dollars), gave
the results:
35
57.9
52
38
38
72
102
38
47
56.9
44.5
64.8
38
46.3
45
53.8
35
38
38
68
54.8
51
44
48
We must first order the data from smallest to largest. I’ll do that for you to save some tedium…
35
47
35
48
38
51
38
52
38
53.8
38
54.8
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38
56.9
38
57.9
44
64.8
44.5
68
1. If someone has an income at the 50 percentile, what was her actual income?
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2. What is the 80 percentile for this data?
45
72
46.3
102
Z-Scores
Z-scores are related to standard deviation. Let’s go back to the data we used for yesterday’s discussion:
50, 55, 53, 62, 60, 52, 48, 51, 58, 53, 47, 50
We found that the range was
62 ! 47 = 15 and that the standard deviation was approximately 4.69.
Since the standard deviation is large compared to the range, it indicates that the data is widely dispersed
around the mean of 53.25. This was confirmed when we looked at the data in graphical form:
A z-score is the number of standard deviations that a single data value (or datum) is from the mean.
To calculate the z-score for a datum, divide its distance from the mean by the standard deviation:
z=
x!x
s
For example, the z-score for the data value 48 is:
48 ! 53.25
4.69
! !1.12
z=
So, the datum 48 is about -1.12 standard deviations from the mean. We can see this visually in the
image above as the datum 48 is just to the left of -1 standard deviations.
What is the z-score for the datum of 60? Make a prediction, then calculate:
z=
This illustrates one final premise. Data values above the mean have positive z-scores, and data values
below the mean have negative z-scores. A datum that is exactly the same as the mean will have a
z-score of 0.