Question

what is the best frequency table for these data?
a.
| pet survey |

number of pets	tally	frequency
0	❂❂❂	15
1	❂❂	10
2	❂		6
3	❁	4
4	❁	3
5	❁	2

b.
| pet survey |

number of pets	tally	frequency
15	❂❂❂	0
10	❂❂	1
6	❂		2
4	❁	3
3	❁	4
2	❁	5

c.
| pet survey |

number of pets	tally	frequency
0	❂❂	10
1	❂❂❂	15
2	❁	4
3	❂		6
4	❁	2
5	❁	3

d.
| pet survey |

number of pets	tally	frequency
10	❂❂	0
15	❂❂❂	1
4	❁	2
6	❂		3
2	❁	4
3	❁	5

Explanation:

Step1: Analyze Tally-Frequency Relationship

A tally mark group (||||) is 5, and individual marks add. For "0 pets", tally should match frequency. Let's check each option:

- Option a: Tally for 0 (three groups of 5: |||| |||| ||||) is 15, frequency 15. Tally for 1 (two groups of 5: |||| ||||) is 10, frequency 10. Tally for 2 (one group of 5 + 1: |||| |) is 6, frequency 6. Tally for 3 (four marks: ||||) is 4, frequency 4. Tally for 4 (three marks: |||) is 3, frequency 3. Tally for 5 (two marks: ||) is 2, frequency 2. All tally-freq match.

- Option b: "Number of Pets" values (15,10,6,4,3,2) don't make sense (should be 0,1,2,3,4,5 for pet count), and tally-freq mismatch (e.g., 15 has tally but frequency 0).

- Option c: Tally for 0 (two groups of 5: |||| ||||) is 10, frequency 10 (correct), but tally for 1 (three groups of 5: |||| |||| ||||) is 15, frequency 15 (correct), but then tally for 2 (four marks: ||||) is 4, frequency 4 (correct), but tally for 3 (one group of 5 + 1: |||| |) is 6, frequency 6 (correct), but tally for 4 (two marks: ||) is 2, frequency 2 (correct), tally for 5 (three marks: |||) is 3, frequency 3 (correct). Wait, but the "Number of Pets" order: 0,1,2,3,4,5 is correct, but tally for 0 is 10 (two groups), but in option a, 0 has 15 (three groups). Wait, no—wait, the key is that the "Number of Pets" should be 0,1,2,3,4,5 (possible pet counts), and tally should match frequency. Wait, no, let's re-express:

Wait, in a frequency table for "Number of Pets", the categories are 0,1,2,3,4,5 (number of pets a person has). So "Number of Pets" column should be 0,1,2,3,4,5 (not 15,10, etc. as in b and d). So b and d are invalid because their "Number of Pets" column has wrong values (15,10,6, etc., which are frequencies, not pet counts). Now between a and c:

In option a:
- 0 pets: tally is three groups of 5 (|||| |||| ||||) → 15, frequency 15.
- 1 pet: two groups of 5 (|||| ||||) → 10, frequency 10.
- 2 pets: one group of 5 + 1 (|||| |) → 6, frequency 6.
- 3 pets: four marks (||||) → 4, frequency 4.
- 4 pets: three marks (|||) → 3, frequency 3.
- 5 pets: two marks (||) → 2, frequency 2.

In option c:
- 0 pets: two groups of 5 (|||| ||||) → 10, frequency 10.
- 1 pet: three groups of 5 (|||| |||| ||||) → 15, frequency 15.
- 2 pets: four marks (||||) → 4, frequency 4.
- 3 pets: one group of 5 + 1 (|||| |) → 6, frequency 6.
- 4 pets: two marks (||) → 2, frequency 2.
- 5 pets: three marks (|||) → 3, frequency 3.

But the "Number of Pets" column in a and c is correct (0,1,2,3,4,5). Now, the tally for each number of pets should correspond. Wait, but in a frequency table, the "Number of Pets" is the category (0 pets, 1 pet, etc.), and the tally is the count of people with that number of pets. So for example, if 15 people have 0 pets, the tally for 0 should be 15 (three groups of 5). If 10 people have 1 pet, tally for 1 is 10 (two groups of 5). If 6 people have 2 pets, tally is 6 (one group of 5 + 1). If 4 people have 3 pets, tally is 4 (four marks). If 3 people have 4 pets, tally is 3 (three marks). If 2 people have 5 pets, tally is 2 (two marks). This matches option a. In option c, the tally for 0 is 10 (two groups) but frequency is 10, tally for 1 is 15 (three groups) but frequency is 15—wait, no, that's correct, but the order of "Number of Pets" and their tallies: in a, 0 has 15, 1 has 10; in c, 0 has 10, 1 has 15. But the key is that the "Number of Pets" column should list the number of pets (0,1,2,3,4,5) in order, and the tally should match the frequency. But also, the "Number of Pets" values in b and d are wrong (they use frequencies as pet counts), so b and d are out. Now between a and c: in a, the tally for 0 is 15 (three groups), frequency 15; tally for 1 is 10 (two groups), frequency 10. In c, tally for 0 is 10 (two groups), frequency 10; tally for 1 is 15 (three groups), frequency 15. But the "Number of Pets" is 0,1,2,3,4,5—so the categories are correct, but the tallies for each category: let's check the tally marks. For 0 pets in a: |||| |||| |||| (15), frequency 15. For 1 pet: |||| |||| (10), frequency 10. For 2 pet: |||| | (6), frequency 6. For 3 pet: |||| (4), frequency 4. For 4 pet: ||| (3), frequency 3. For 5 pet: || (2), frequency 2. All tally counts equal frequency. In c, for 0 pet: |||| |||| (10), frequency 10 (correct), 1 pet: |||| |||| |||| (15), frequency 15 (correct), but then 2 pet: |||| (4), frequency 4 (correct), 3 pet: |||| | (6), frequency 6 (correct), 4 pet: || (2), frequency 2 (correct), 5 pet: ||| (3), frequency 3 (correct). Wait, but the problem is the "Number of Pets" column: in a, the numbers are 0,1,2,3,4,5 (correct), and the tallies match frequencies. In c, same numbers, but the tallies for 0 and 1 are swapped? Wait no, the "Number of Pets" is the category (how many pets), so 0 pets: people with 0 pets, 1 pet: people with 1 pet, etc. So the tally for 0 pets is the number of people with 0 pets, which is 15 in a, 10 in c. But the key is that in a frequency table, the "Number of Pets" should be the possible values (0,1,2,3,4,5), and the tally should correspond to the frequency (number of people with that many pets). Now, looking at the options, b and d have "Number of Pets" as 15,10,6, etc., which are frequencies, not pet counts, so they are invalid. Now between a and c: in a, the "Number of Pets" is 0,1,2,3,4,5 (correct), and each tally's count (sum of tally marks) equals the frequency. Let's verify a:

- 0 pets: tally is three groups of 5 (|||| |||| ||||) → 5×3=15, frequency 15 ✔️
- 1 pet: two groups of 5 (|||| ||||) → 5×2=10, frequency 10 ✔️
- 2 pets: one group of 5 + 1 (|||| |) → 5+1=6, frequency 6 ✔️
- 3 pets: four marks (||||) → 4, frequency 4 ✔️
- 4 pets: three marks (|||) → 3, frequency 3 ✔️
- 5 pets: two marks (||) → 2, frequency 2 ✔️

In c:

- 0 pets: two groups of 5 (|||| ||||) → 10, frequency 10 ✔️
- 1 pet: three groups of 5 (|||| |||| ||||) → 15, frequency 15 ✔️
- 2 pets: four marks (||||) → 4, frequency 4 ✔️
- 3 pets: one group of 5 + 1 (|||| |) → 6, frequency 6 ✔️
- 4 pets: two marks (||) → 2, frequency 2 ✔️
- 5 pets: three marks (|||) → 3, frequency 3 ✔️

Wait, but then why is a the best? Wait, maybe the original data (not shown) has 15 people with 0 pets, 10 with 1, 6 with 2, 4 with 3, 3 with 4, 2 with 5. So the frequency table should have "Number of Pets" as 0,1,2,3,4,5 (the number of pets), and the frequency (number of people) matching the tally. In option a, the "Number of Pets" is 0,1,2,3,4,5 (correct), and the tally for each matches the frequency. In option c, the "Number of Pets" is 0,1,2,3,4,5, but the tally for 0 is 10 (frequency 10) and 1 is 15 (frequency 15), which would mean 10 people have 0 pets and 15 have 1, but maybe the original data is 15 with 0, 10 with 1. Also, in option a, the "Number of Pets" column is in ascending order (0,1,2,3,4,5), which is standard for frequency tables. Option c also has ascending order, but the tallies for 0 and 1 are swapped in terms of count. But the key mistake in c, b, d: b and d have "Number of Pets" as 15,10, etc., which are frequencies, not pet counts. So b and d are invalid. Between a and c, a has the correct tally-freq match for each "Number of Pets" (0,1,2,3,4,5) with the tallies corresponding to the number of people (frequency) for each pet count. So a is correct.

Answer:

a. Pet Survey

Number of Pets	Tally	Frequency
1										10
2							6
3						4
4					3
5				2