Question

calculate the 75th percentile of the data shown
x
3.6
3.9
8.6
11
11.4
11.7
13.9
21.7
22.3
24.2

Explanation:

Step1: Determine the number of data points

The data set has \( n = 10 \) values.

Step2: Calculate the position of the 75th percentile

The formula for the position \( i \) of the \( p \)-th percentile is \( i=\frac{p}{100}(n + 1) \). For \( p = 75 \) and \( n = 10 \), we have \( i=\frac{75}{100}(10 + 1)=\frac{3}{4}\times11 = 8.25 \).

Step3: Interpolate to find the 75th percentile

The 8th value in the ordered data set is \( 21.7 \) and the 9th value is \( 22.3 \). The decimal part of \( i = 8.25 \) is \( 0.25 \). We use the formula: \( \text{Percentile}=x_{8}+0.25\times(x_{9}-x_{8}) \). Substituting the values, we get \( 21.7+0.25\times(22.3 - 21.7)=21.7 + 0.25\times0.6=21.7 + 0.15 = 21.85 \). Wait, no, wait. Wait, actually, another common method for percentiles (especially in some textbooks) is using \( i=\frac{p}{100}n \). Let's check that. If \( i=\frac{75}{100}\times10 = 7.5 \). Then, the 7.5th percentile. The 7th value is \( 13.9 \), 8th is \( 21.7 \). Wait, no, I think I ordered the data wrong? Wait, the data is: 3.6, 3.9, 8.6, 11, 11.4, 11.7, 13.9, 21.7, 22.3, 24.2. So ordered from least to greatest, that's correct. Wait, maybe the first method was wrong. Let's recall the two common methods:

Method 1 (used in some stats courses): \( i=\frac{p}{100}(n - 1)+1 \). So \( i=\frac{75}{100}(10 - 1)+1=0.75\times9 + 1 = 6.75 + 1 = 7.75 \). Wait, no, different sources have different methods. Wait, the most common method for percentiles (especially when \( n \) is the number of observations) is:

The position \( i \) is calculated as \( i=\frac{p}{100}\times n \). If \( i \) is an integer, the percentile is the average of the \( i \)-th and \( (i + 1) \)-th values. If \( i \) is not an integer, we take the next integer greater than \( i \) and the value at that position? No, wait, let's check the correct method.

Wait, the data set has \( n = 10 \) observations. To find the 75th percentile:

1. Order the data: 3.6, 3.9, 8.6, 11, 11.4, 11.7, 13.9, 21.7, 22.3, 24.2 (already ordered).

2. Calculate the index \( i=\frac{p}{100}\times n=\frac{75}{100}\times10 = 7.5 \).

Since \( i = 7.5 \) is not an integer, we interpolate between the 7th and 8th values. The 7th value is \( 13.9 \), the 8th value is \( 21.7 \)? Wait, no, wait the 7th value: let's count:

1: 3.6

2: 3.9

3: 8.6

4: 11

5: 11.4

6: 11.7

7: 13.9

8: 21.7

9: 22.3

10: 24.2

Ah, here's the mistake earlier. So the 7th value is 13.9, 8th is 21.7? Wait, no, that can't be, there's a big jump. Wait, no, the data is: 3.6, 3.9, 8.6, 11, 11.4, 11.7, 13.9, 21.7, 22.3, 24.2. So between 13.9 and 21.7 is a big gap. But let's use the correct formula for percentiles. The formula for the \( p \)-th percentile (where \( p \) is the percentile, \( n \) is the number of observations) is:

\( i=\frac{p}{100}(n + 1) \)

So for \( p = 75 \), \( n = 10 \):

\( i=\frac{75}{100}(10 + 1)=8.25 \)

So the 8.25th percentile. The 8th value is 21.7, the 9th value is 22.3.

To interpolate, we take the 8th value plus 0.25 times the difference between the 9th and 8th values.

So:

\( \text{75th percentile}=21.7+0.25\times(22.3 - 21.7) \)

\( =21.7+0.25\times0.6 \)

\( =21.7 + 0.15 \)

\( =21.85 \)

Wait, but that seems odd because the 75th percentile should be a value that 75% of the data is below it. Let's check the number of data points below 21.85. The data points are: 3.6, 3.9, 8.6, 11, 11.4, 11.7, 13.9, 21.7 (that's 8 points) and 22.3, 24.2 (2 points). Wait, 8 points are below 21.85? Wait, 21.7 is the 8th point, 22.3 is the 9th. So 8 points are ≤21.7, 2 points are >21.7. Wait, maybe the formula is different. Let's check another method. The percentile is the value where at least \( p\% \) of the data is less than or equal to it and at least \( (100 - p)\% \) is greater than or equal to it.

For \( n = 10 \), 75% of 10 is 7.5. So we need the value such that 7.5 data points are below or equal to it. So we take the 8th value? Wait, no. Let's use the formula from the textbook:

The position \( i \) is calculated as \( i=\frac{p}{100}\times n \). If \( i \) is an integer, the percentile is the average of the \( i \)-th and \( (i + 1) \)-th observations. If \( i \) is not an integer, we round up to the next integer, and the percentile is the observation at that position. Wait, no, different sources have different methods. Let's check the data again.

Wait, the data set is: 3.6, 3.9, 8.6, 11, 11.4, 11.7, 13.9, 21.7, 22.3, 24.2. So ordered.

Number of observations \( n = 10 \).

To find the 75th percentile:

1. Calculate \( i=\frac{75}{100}\times10 = 7.5 \).

Since \( i \) is not an integer, we use linear interpolation between the 7th and 8th observations.

The 7th observation is 13.9, the 8th is 21.7.

The difference between the 8th and 7th is \( 21.7 - 13.9 = 7.8 \).

The fractional part of \( i \) is 0.5, so we add 0.5 times the difference to the 7th observation:

\( 13.9+0.5\times7.8 = 13.9 + 3.9 = 17.8 \). Wait, that's different. Now I'm confused. Which method is correct?

Wait, let's check the definition of percentile. The 75th percentile (also called the third quartile) is the value where 75% of the data is less than or equal to it.

For a sample with \( n \) observations, the formula for the position of the \( p \)-th percentile is:

\( i=\frac{p}{100}(n - 1)+1 \)

Let's use this formula. So \( i=\frac{75}{100}(10 - 1)+1 = 0.75\times9 + 1 = 6.75 + 1 = 7.75 \).

So the 7.75th position. The 7th observation is 13.9, the 8th is 21.7.

The difference between the 8th and 7th is \( 21.7 - 13.9 = 7.8 \).

The fractional part is 0.75, so we add 0.75 times the difference to the 7th observation:

\( 13.9+0.75\times7.8 = 13.9 + 5.85 = 19.75 \). No, that's not matching.

Wait, maybe I made a mistake in the data order. Wait, the data is: 3.6, 3.9, 8.6, 11, 11.4, 11.7, 13.9, 21.7, 22.3, 24.2. So that's 10 data points. Let's count how many are below a certain value.

We need 75% of 10, which is 7.5, so at least 7.5 data points below the percentile. So the 8th data point? Wait, 8 data points are below or equal to 21.7 (since the 8th is 21.7). Wait, 3.6, 3.9, 8.6, 11, 11.4, 11.7, 13.9, 21.7: that's 8 data points. Then 22.3 and 24.2 are above. So 8 out of 10 is 80%, which is more than 75%. Wait, maybe the correct method is to use the formula \( i=\frac{p}{100}(n + 1) \). So \( i = 8.25 \), as I first calculated. So the 8.25th data point. The 8th is 21.7, the 9th is 22.3. So 21.7 + 0.25*(22.3 - 21.7) = 21.7 + 0.15 = 21.85. Let's check how many data points are below 21.85. The 8th data point is 21.7, which is below 21.85, and the 9th is 22.3, which is above. So 8 data points are below 21.85, which is 80% of 10, but we need 75%. Wait, maybe the data set is small, and the correct percentile is the 8th value? But 8th value is 21.7, and 8/10 = 80% ≤ 75%? No, 80% is more than 75%. Wait, maybe the formula is \( i=\frac{p}{100}n \), and if \( i \) is not an integer, we take the ceiling of \( i \). So \( i = 7.5 \), ceiling is 8, so the 8th value is 21.7. But 8/10 = 80% of the data is below or equal to 21.7, which is more than 75%. Alternatively, maybe the 7th value is 13.9, 7/10 = 70% ≤75%, 8/10=80%>75%, so the 75th percentile is between the 7th and 8th values. So using linear interpolation between the 7th (13.9) and 8th (21.7) values. The fraction is (75 - 70)/(80 - 70) = 5/10 = 0.5. So the percentile is 13.9 + 0.5*(21.7 - 13.9) = 13.9 + 3.9 = 17.8. But that seems odd. Wait, maybe I messed up the data. Wait, the data is: 3.6, 3.9, 8.6, 11, 11.4, 11.7, 13.9, 21.7, 22.3, 24.2. Let's sort them again:

1: 3.6

2: 3.9

3: 8.6

4: 11

5: 11.4

6: 11.7

7: 13.9

8: 21.7

9: 22.3

10: 24.2

Yes, that's correct. Now, let's use the formula from the National Institute of Standards and Technology (NIST) for percentiles:

The \( p \)-th percentile of a data set with \( n \) observations is the value such that at least \( p\% \) of the observations are less than or equal to it and at least \( (100 - p)\% \) are greater than or equal to it.

To calculate the position \( i \):

\( i=\frac{p}{100}(n + 1) \)

For \( p = 75 \), \( n = 10 \):

\( i=\frac{75}{100}(10 + 1)=8.25 \)

So the 8.25th observation. This means we take the 8th observation plus 0.25 times the difference between the 9th and 8th observations.

8th observation: 21.7

9th observation: 22.3

Difference: 22.3 - 21.7 = 0.6

So 21.7 + 0.25*0.6 = 21.7 + 0.15 = 21.85

So the 75th percentile is 21.85.

Answer:

21.85