Question

1. for the following set of scores:

12 13 8 14 10 8 9 13 9
9 14 8 12 8 13 13 7 12
a. organize the scores in a frequency distribution table.
b. based on the frequencies, identify the shape of the distribution.

Explanation:

Part (a)

Step 1: Identify unique scores

First, we list out the unique scores from the given data set: 7, 8, 9, 10, 12, 13, 14.

Step 2: Count frequencies

Now we count how many times each score appears:

• Score 7: appears 1 time.
• Score 8: let's count, the 8s are at positions 3, 6, 13, 15. So 4 times.
• Score 9: positions 7, 10, 11? Wait, original data: 12, 13, 8, 14, 10, 8, 9, 13, 9, 9, 14, 8, 12, 8, 13, 13, 7, 12. Wait, let's list all scores: 12,13,8,14,10,8,9,13,9,9,14,8,12,8,13,13,7,12. Let's count each:
• 7: 1 (only the 17th score)
• 8: scores at index 2 (8), 5 (8), 11 (8), 13 (8) → 4 times? Wait no, let's count again: 8 appears at positions 3 (8), 6 (8), 12 (8), 14 (8)? Wait the data is: 12, 13, 8, 14, 10, 8, 9, 13, 9, 9, 14, 8, 12, 8, 13, 13, 7, 12. So the 8s are at 3rd, 6th, 12th, 14th → that's 4? Wait no, 3rd:8, 6th:8, 12th:8, 14th:8 → 4? Wait no, wait the data has 18 scores? Wait 12,13,8,14,10,8,9,13,9,9,14,8,12,8,13,13,7,12 → let's count the number of elements: 1 (12),2(13),3(8),4(14),5(10),6(8),7(9),8(13),9(9),10(9),11(14),12(8),13(12),14(8),15(13),16(13),17(7),18(12). So total 18 scores. Now count 8: positions 3,6,12,14 → 4? Wait no, 3,6,12,14: that's 4? Wait 8 appears 4 times? Wait no, let's list all 8s: 8 (3), 8 (6), 8 (12), 8 (14) → 4? Wait no, maybe I missed. Wait the data: 12,13,8,14,10,8,9,13,9,9,14,8,12,8,13,13,7,12. So 8s are at index 2 (if 0-based) or 3 (1-based). Wait 1-based: 3,6,12,14 → four 8s? Wait no, 3:8, 6:8, 12:8, 14:8 → that's four. Then 9: positions 7,9,10 → 7(9),9(9),10(9) → three? Wait no, 7th score is 9, 9th is 9, 10th is 9 → three? Wait no, 7,9,10: three 9s? Wait no, 7:9, 9:9, 10:9 → three? Wait the 10th score is 9, 9th is 9, 7th is 9 → three? Wait no, let's count again: 9 appears at 7,9,10 → three? Wait no, 7:9, 9:9, 10:9 → three? Wait the data: 12,13,8,14,10,8,9,13,9,9,14,8,12,8,13,13,7,12. So 9s: 7th (9), 9th (9), 10th (9) → three? Wait no, 7,9,10: three 9s? Wait 7:9, 9:9, 10:9 → three. Then 10: appears once (5th score:10). 12: positions 1,13,18 → 1(12),13(12),18(12) → three times. 13: positions 2,8,15,16 → 2(13),8(13),15(13),16(13) → four times. 14: positions 4,11 → 4(14),11(14) → two times. 7: once (17th). Wait let's tabulate:

Score (X) | Frequency (f)
--- | ---
7 | 1
8 | 4 (scores at 3,6,12,14)
9 | 3 (scores at 7,9,10)
10 | 1 (score at 5)
12 | 3 (scores at 1,13,18)
13 | 4 (scores at 2,8,15,16)
14 | 2 (scores at 4,11)

Wait wait, let's check the count of total frequencies: 1+4+3+1+3+4+2 = 18, which matches the number of scores (18). Good.

Step 3: Create frequency table

We can also present it in a table with columns for Score (X), Frequency (f), and optionally Relative Frequency, Cumulative Frequency, etc. But the question just asks for frequency distribution table, so:

Score (X)	Frequency (f)
8	4
9	3
10	1
12	3
13	4
14	2

Part (b)

To identify the shape of the distribution, we look at the frequencies. A distribution is symmetric if the frequencies are mirrored around the center, skewed left if tail is on the left, skewed right if tail is on the right. Let's list the scores in order: 7,8,8,8,8,9,9,9,10,12,12,12,13,13,13,13,14,14. Wait wait, no, when we order the scores: 7,8,8,8,8,9,9,9,10,12,12,12,13,13,13,13,14,14. Wait the middle is between the 9th and 10th scores. 9th score is 10, 10th is 12. Wait but looking at frequencies: scores 8 and 13 both have frequency 4, 9 and 12 both have frequency 3, 7 and 10 both have frequency 1, 14 has frequency 2. Wait the left side (lower scores:7,8,9,10) and right side (higher scores:12,13,14) – let's check the frequencies:

For lower scores (7,8,9,10): frequencies 1,4,3,1. Sum: 1+4+3+1=9.

For higher scores (12,13,14): frequencies 3,4,2. Sum: 3+4+2=9. Wait, that's symmetric? Wait but 10 is on the left, 12 on the right. Wait the frequency of 8 (4) and 13 (4), 9 (3) and 12 (3), 7 (1) and... 14 has frequency 2, 10 has 1. Wait maybe I made a mistake in ordering. Wait original scores: 7,8,8,8,8,9,9,9,10,12,12,12,13,13,13,13,14,14. Wait the number of scores is 18, so median is between 9th and 10th. 9th score:10, 10th:12. Now, the mode: scores with highest frequency are 8 and 13 (both f=4). So bimodal? Wait no, the question is about the shape (symmetric, skewed left, skewed right). Let's check the frequencies around the center. The middle scores are around 10-12. The left side (below median) has scores 7,8,9,10 with frequencies 1,4,3,1. The right side (above median) has 12,13,14 with frequencies 3,4,2. Wait the frequencies for 8 (4) and 13 (4), 9 (3) and 12 (3), 7 (1) and 14 (2) – close. Alternatively, maybe it's symmetric because the frequencies of 8 and 13 are equal, 9 and 12 are equal, and the lower and higher tails (7 and 14) have frequencies 1 and 2, which are close. Wait maybe the distribution is symmetric (or approximately symmetric) because the frequencies on either side of the center (around 10-12) are mirroring each other (8 and 13 both 4, 9 and 12 both 3, 7 and 14 close in frequency). Alternatively, maybe slightly skewed, but given the frequencies, 8 and 13 have same frequency, 9 and 12 same, so it's symmetric.

Part (a) Answer:

The frequency distribution table is:

Score (X)	Frequency (f)
8	4
9	3
10	1
12	3
13	4
14	2

Part (b) Answer:

The distribution is approximately symmetric (since the frequencies of corresponding scores on either side of the center are similar: 8 and 13 both have frequency 4, 9 and 12 both have frequency 3, and the lower and upper tails have relatively close frequencies).

Answer:

To identify the shape of the distribution, we look at the frequencies. A distribution is symmetric if the frequencies are mirrored around the center, skewed left if tail is on the left, skewed right if tail is on the right. Let's list the scores in order: 7,8,8,8,8,9,9,9,10,12,12,12,13,13,13,13,14,14. Wait wait, no, when we order the scores: 7,8,8,8,8,9,9,9,10,12,12,12,13,13,13,13,14,14. Wait the middle is between the 9th and 10th scores. 9th score is 10, 10th is 12. Wait but looking at frequencies: scores 8 and 13 both have frequency 4, 9 and 12 both have frequency 3, 7 and 10 both have frequency 1, 14 has frequency 2. Wait the left side (lower scores:7,8,9,10) and right side (higher scores:12,13,14) – let's check the frequencies:

For lower scores (7,8,9,10): frequencies 1,4,3,1. Sum: 1+4+3+1=9.

For higher scores (12,13,14): frequencies 3,4,2. Sum: 3+4+2=9. Wait, that's symmetric? Wait but 10 is on the left, 12 on the right. Wait the frequency of 8 (4) and 13 (4), 9 (3) and 12 (3), 7 (1) and... 14 has frequency 2, 10 has 1. Wait maybe I made a mistake in ordering. Wait original scores: 7,8,8,8,8,9,9,9,10,12,12,12,13,13,13,13,14,14. Wait the number of scores is 18, so median is between 9th and 10th. 9th score:10, 10th:12. Now, the mode: scores with highest frequency are 8 and 13 (both f=4). So bimodal? Wait no, the question is about the shape (symmetric, skewed left, skewed right). Let's check the frequencies around the center. The middle scores are around 10-12. The left side (below median) has scores 7,8,9,10 with frequencies 1,4,3,1. The right side (above median) has 12,13,14 with frequencies 3,4,2. Wait the frequencies for 8 (4) and 13 (4), 9 (3) and 12 (3), 7 (1) and 14 (2) – close. Alternatively, maybe it's symmetric because the frequencies of 8 and 13 are equal, 9 and 12 are equal, and the lower and higher tails (7 and 14) have frequencies 1 and 2, which are close. Wait maybe the distribution is symmetric (or approximately symmetric) because the frequencies on either side of the center (around 10-12) are mirroring each other (8 and 13 both 4, 9 and 12 both 3, 7 and 14 close in frequency). Alternatively, maybe slightly skewed, but given the frequencies, 8 and 13 have same frequency, 9 and 12 same, so it's symmetric.

Part (a) Answer:

The frequency distribution table is:

Score (X)	Frequency (f)
8	4
9	3
10	1
12	3
13	4
14	2

Part (b) Answer:

The distribution is approximately symmetric (since the frequencies of corresponding scores on either side of the center are similar: 8 and 13 both have frequency 4, 9 and 12 both have frequency 3, and the lower and upper tails have relatively close frequencies).