Question

the data set below represents the ages of 36 executives. find the percentile that corresponds to an age of 51 years old.
28 29 29 29 32 35 38 39 39 41
41 41 44 44 44 45 46 47 49 49
51 51 53 54 54 55 56 57 58 61
62 63 63 64 64 64

percentile of 51 = \\(\square\\) (round to the nearest integer as needed.)

Explanation:

Step1: Count values less than 51

First, we count how many ages in the data set are less than 51. Let's list out the ages:
First row: 28, 29, 29, 29, 32, 35, 38, 39, 39, 41
Second row: 41, 41, 44, 44, 44, 45, 46, 47, 49, 49
Third row: 51, 51, 53, 54, 54, 55, 56, 57, 58, 61
Fourth row: 62, 63, 63, 64, 64, 64

Now, count the numbers less than 51. Let's count:
First row: 10 numbers (28,29,29,29,32,35,38,39,39,41)
Second row: 10 numbers (41,41,44,44,44,45,46,47,49,49)
Wait, wait, let's check again. Wait, the third row starts with 51. So total numbers less than 51: let's count each number:

First row: 28,29,29,29,32,35,38,39,39,41 → 10 numbers
Second row: 41,41,44,44,44,45,46,47,49,49 → 10 numbers
Wait, but 41 is less than 51, 44,45,46,47,49 are all less than 51. So second row is 10 numbers. Then, is there any number between second row and third row? Wait, the data set has 36 numbers. Let's count the total numbers:

First row: 10
Second row: 10 (total 20)
Third row: 10 (total 30)
Fourth row: 6 (total 36). Wait, maybe I miscounted. Let's count again:

First row: 28,29,29,29,32,35,38,39,39,41 → 10
Second row: 41,41,44,44,44,45,46,47,49,49 → 10 (total 20)
Third row: 51,51,53,54,54,55,56,57,58,61 → 10 (total 30)
Fourth row: 62,63,63,64,64,64 → 6 (total 36). Wait, that's 10+10+10+6=36. So the numbers less than 51: first row (10) + second row (10) = 20? Wait, no. Wait, 51 is in the third row. So numbers less than 51: let's list all numbers and count:

Numbers:
28,29,29,29,32,35,38,39,39,41,
41,41,44,44,44,45,46,47,49,49,
51,51,53,54,54,55,56,57,58,61,
62,63,63,64,64,64

Now, count how many are less than 51:

First 20 numbers (first two rows) are all less than 51? Wait, 49 is less than 51, yes. Then the third row starts with 51. Wait, the third row's first number is 51. So numbers less than 51: first 20 numbers? Wait, no. Wait, let's count:

First row: 10 numbers (all <51)
Second row: 10 numbers (all <51)
Third row: first two numbers are 51, which are not less than 51. So total numbers less than 51: 10 + 10 = 20? Wait, but wait, 51 is in the data set. Wait, the problem is to find the percentile for age 51. The formula for percentile is:

Percentile of a value \( x \) is \( \frac{\text{Number of values less than } x}{\text{Total number of values}} \times 100 \)

Wait, but sometimes it's also calculated as \( \frac{\text{Number of values less than or equal to } x - 0.5}{\text{Total number of values}} \times 100 \), but the standard formula for percentile (specifically, the percentile rank) is the number of values less than \( x \) divided by total number, times 100.

Wait, let's check the data again. Let's list all 36 numbers:

1:28, 2:29, 3:29, 4:29, 5:32, 6:35, 7:38, 8:39, 9:39, 10:41,

11:41, 12:41, 13:44, 14:44, 15:44, 16:45, 17:46, 18:47, 19:49, 20:49,

21:51, 22:51, 23:53, 24:54, 25:54, 26:55, 27:56, 28:57, 29:58, 30:61,

31:62, 32:63, 33:63, 34:64, 35:64, 36:64

Now, count the number of values less than 51. Let's see: positions 1 to 20: all are less than 51 (since position 20 is 49). Then position 21 is 51. So number of values less than 51 is 20.

Total number of values is 36.

So percentile is \( \frac{20}{36} \times 100 \approx 55.56 \). Wait, but wait, maybe we need to consider values less than or equal? Wait, no, the percentile rank is typically the percentage of data points that are less than the given value.

Wait, but let's check the formula again. The formula for the percentile corresponding to a value \( x \) is:

\( \text{Percentile} = \frac{\text{Number of data values less than } x}{\text{Total number of data values}} \times 100 \)

So here, number of data values less than 51: let's count again. The data points are:

28,29,29,29,32,35,38,39,39,41,

41,41,44,44,44,45,46,47,49,49,

51,51,53,54,54,55,56,57,58,61,

62,63,63,64,64,64

So the first 20 data points (positions 1 - 20) are all less than 51. Then positions 21 and 22 are 51. So number of values less than 51 is 20.

Total number of values is 36.

So percentile is \( \frac{20}{36} \times 100 \approx 55.56 \), which rounds to 56? Wait, but wait, maybe I made a mistake in counting. Wait, let's count the number of values less than 51:

First row: 10 numbers (all <51)

Second row: 10 numbers (all <51)

Third row: first two numbers are 51, so the rest in third row (positions 23 - 30) are 53,54,54,55,56,57,58,61 (8 numbers), which are >51.

Fourth row: 6 numbers (62,63,63,64,64,64) >51.

So total less than 51: 10 + 10 = 20.

Total data points: 36.

So \( \frac{20}{36} \times 100 \approx 55.56 \), which rounds to 56? Wait, but let's check another way. Maybe the formula is \( \frac{\text{Number of values less than or equal to } x - 0.5}{\text{Total number of values}} \times 100 \). Let's try that.

Number of values less than or equal to 51: positions 1 - 22 (since positions 21 and 22 are 51). So 22 values.

Then, \( \frac{22 - 0.5}{36} \times 100 = \frac{21.5}{36} \times 100 \approx 59.72 \), which rounds to 60? But that doesn't seem right.

Wait, maybe the correct formula is the number of values less than x divided by total, times 100. Let's check with an example. Suppose we have data [1,2,3,4,5], and we want the percentile of 3. Number of values less than 3: 2 (1,2). Total:5. Percentile: (2/5)*100=40. Which is correct, as 40% of data is less than 3.

So applying that here, number of values less than 51: 20. Total:36. So (20/36)*100 ≈55.56, which rounds to 56. Wait, but let's check the data again. Wait, the third row has two 51s. So the data points are:

1:28

2:29

3:29

4:29

5:32

6:35

7:38

8:39

9:39

10:41

11:41

12:41

13:44

14:44

15:44

16:45

17:46

18:47

19:49

20:49

21:51

22:51

23:53

24:54

25:54

26:55

27:56

28:57

29:58

30:61

31:62

32:63

33:63

34:64

35:64

36:64

So numbers less than 51: positions 1-20, which is 20 numbers. So 20/36=0.555..., times 100 is 55.555..., which rounds to 56. Wait, but maybe the answer is 56? Wait, but let's check another source. The percentile formula is:

Percentile = (Number of values below x / Total number of values) * 100

So here, number of values below 51 is 20, total is 36. So 20/36 ≈0.5556, times 100 is 55.56, which rounds to 56. So the percentile of 51 is 56.

Wait, but let's confirm. Let's count again:

First row: 10 numbers (all <51)

Second row: 10 numbers (all <51)

Third row: first two numbers are 51 (not <51), so 2 numbers.

Fourth row: 6 numbers (not <51)

Total <51: 10+10=20. Total data:36.

20/36=5/9≈0.5555, times 100≈55.56, which rounds to 56.

So the percentile is 56.

Step2: Calculate the percentile

Using the formula for percentile:

\( \text{Percentile} = \frac{\text{Number of values less than } 51}{\text{Total number of values}} \times 100 \)

We have number of values less than 51 = 20, total number of values = 36.

So,

\( \text{Percentile} = \frac{20}{36} \times 100 \approx 55.56 \)

Rounding to the nearest integer, we get 56.

Answer:

56