Estimating Percentiles From The Cumulative Relative Frequency Table
The data above corresponds to the week 6 quiz score data in the course packet.
Complete the relative frequency and cumulative relative frequency columns.
x
Frequency Relative Frequency
2
1
0.05
5
1
0.05
7
1
0.05
10
1
0.05
12
2
0.10
13
1
0.05
14
2
0.10
15
3
0.15
16
1
0.05
17
2
0.10
18
1
0.05
20
4
0.20
20
1
Cumulative
Relative Frequency
0.05
0.10
0.15
0.20
0.30
0.35
0.45
0.60
0.65
0.75
0.80
1.00
To estimate the pth percentile, move down the cumulative relative frequency column until the first
line at which you find or pass the value of p for the percentile you are looking for.
If you pass beyond the value of p that you are looking for, the value for the percentile is the x value
in the data column at the first line beyond the percentile you were looking for.
To find the 40th percentile
Go down right hand column looking for 0.40.
You don’t find it exactly. You pass it between 0.35 and 0.45.
Then the 40th percentile is the x value for the line at which you first pass 0.40
The 40th percentile is 14
If you find the exact value of p that you are looking for, the value for the percentile is the average of
the x value in that line and the x value in the next line.
To find the 30th percentile
Go down right hand column looking for 0.30.
You find 0.30 on the line where x = 12.
You estimate that the 30th percentile is about 12.
To be technically correct, the 30th percentile would then be the average of the x value on that line (12) and
the x value on the next line (13).
The 30th percentile is (12+13)/2=12.5
TRY IT!
Find the 70 th percentile and the 80th percentile.
Don’t look at the next page of this document until you have tried it. Then read page 2 to see the answers.
More Percentile Practice:
Find the 20 th percentile and the 25th percentile (answers on page 3 of this document)
Write the sentences that interpret these percentiles in the context of the situation in this problem.
Quartile Practice:
Find the 25th and 75th percentiles using this method.
Find the first and third quartiles using your calculator.
Do they agree with each other? (see page 4 of this document for the answers)
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To estimate the pth percentile:
Construct the cumulative relative frequency column in the table.
To estimate the pth percentile, move down the cumulative relative frequency column until the first
line at which you find or pass the value of p for the percentile you are looking for.
If you pass beyond the value of p that you are looking for, the value for the percentile is the x value
in the data column at the first line beyond the percentile you were looking for.
If you find the exact value of p that you are looking for, the value for the percentile is the average of
the x value in that line and the x value in the next line.
To find the 70th percentile
Go down right hand column looking for 0.70.
You don’t find it exactly. You pass it between 0.65 and 0.75.
Then the 70th percentile is the x value for the line at which you first go beyond 0.70
The 70th percentile is 17.
Interpretation: 70% of students had quiz scores of 17 or less.
To find the 80th percentile
Go down right hand column looking for 0.80.
You find 0.80 on the line where x = 18.
You estimate that the 80th percentile is about 18.
To be technically correct, the 80th percentile would then be the average of the x value on that line (18) and
the x value on the next line (20).
The 80th percentile is (18+20)/2=19.
Interpretation: 80 % of students had quiz scores of 19 or less
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To estimate the pth percentile:
Construct the cumulative relative frequency column in the table.
To estimate the pth percentile, move down the cumulative relative frequency column until the first
line at which you find or pass the value of p for the percentile you are looking for.
If you pass beyond the value of p that you are looking for, the value for the percentile is the x value
in the data column at the first line beyond the percentile you were looking for.
If you find the exact value of p that you are looking for, the value for the percentile is the average of
the x value in that line and the x value in the next line.
To find the 20th percentile
Go down right hand column looking for 0.20.
You find 0.20 on the line where x = 10.
You estimate that the 20th percentile is about 10.
To be technically correct, the 20th percentile would then be the average of the x value on that line (10) and
the x value on the next line (12).
The 20th percentile is (10+12)/2=11.
Interpretation: 20 % of students had quiz scores of 11 or less
To find the 25th percentile
Go down right hand column looking for 0.25.
You don’t find it exactly. You pass it between 0.20 and 0.30
Then the 25th percentile is the x value for the line at which you first go beyond 0.25
The 25th percentile is 12.
Interpretation: 25% of students had quiz scores of 12 or less.
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To find quartiles Q1 and Q3 you are looking for the 25th and 75th percentiles
To estimate the pth percentile:
Construct the cumulative relative frequency column in the table.
To estimate the pth percentile, move down the cumulative relative frequency column until the first
line at which you find or pass the value of p for the percentile you are looking for.
If you pass beyond the value of p that you are looking for, the value for the percentile is the x value
in the data column at the first line beyond the percentile you were looking for.
If you find the exact value of p that you are looking for, the value for the percentile is the average of
the x value in that line and the x value in the next line.
To find the 25th percentile
Go down right hand column looking for 0.25.
You don’t find it exactly. You pass it between 0.20 and 0.30
Then the 25th percentile is the x value for the line at which you first go beyond 0.25
The 25th percentile is 12.
To find the 75th percentile
Go down right hand column looking for 0.75.
You find 0.75 on the line where x = 17.
You estimate that the 75th percentile is about 17.
To be technically correct, the 75th percentile would then be the average of the x value on that line (17) and
the x value on the next line (18).
The 75th percentile is (17+18)/2=17.5
Use your calculator to find the quartiles using one variable statistics
2
5
7
10 12 12 13 14 14 15 15 15 16 17 17 18 20 20 20 20
Q1=12
½ (12+12)
Median=15
½ (15+15)
Q3 =17.5
½ (17 + 18)
The TI calculators find the lower quartile by finding the median of the lower half of the data and
the upper quartile by finding the median of the upper half of the data.
In this case your calculator will find Q1 = 12 and Q3 = 17.5
The cumulative relative frequency method and your calculator’s method gave the same result.
Be aware that:
When using different technology tools to do statistical analysis you may find that sometimes
different technology will give different answers. Excel, your calculator, the cumulative relative
frequency method by hand, the locator method in the textbook, and statistical software packages may give
some answers that are the same and some that are close but not exactly the same. This is because the
different technologies use different programs that may round differently to find the position of the
percentile or quartile. Percentiles are most appropriately used with very large data sets.
Then there are usually not large gaps between the data values, and the different methods give more
consistent answers for quartiles or for percentiles. Because we are working with small data sets with gaps
between the data values, we find more inconsistencies between the various methods.
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