what is the center of the data?\nnumber of laps around a track\nthe center is \\boxed{}.

Answer

Explanation:

Step1: Count data points per lap

Lap 1: 8 points, Lap 2: 6 points, Lap 3: 4 points, Lap 4: 2 points, Lap 5: 1 point. Total points: (8 + 6 + 4 + 2 + 1 = 21).

Step2: Find median position

Median is at (\frac{21 + 1}{2} = 11)-th position (since median for odd (n) is (\frac{n + 1}{2})-th term).

Step3: Cumulative count to find median

Lap 1: 8 (positions 1–8)
Lap 2: 8 + 6 = 14 (positions 9–14)
11th position is in Lap 2? Wait, no—wait, lap 1: 8, lap 2: 6 (total 14), lap 3: 4 (total 18), lap 4: 2 (total 20), lap 5:1 (total 21). Wait, 11th term: after lap 1 (8), 11 - 8 = 3rd term in lap 2? No, wait, lap 1 has 8, lap 2 has 6 (so positions 9–14: lap 2). Wait, no—wait, the dot plot: lap 1 (x=1) has 8 dots, lap 2 (x=2) has 6 dots, lap 3 (x=3) has 4, lap 4 (x=2), lap 5 (x=1). Wait, maybe I miscounted. Let's recount dots:

Lap 1 (x=1): Let's count the dots. The vertical dots: 8 (since 8 dots). Lap 2 (x=2): 6 dots. Lap 3 (x=3): 4 dots. Lap 4 (x=4): 2 dots. Lap 5 (x=5): 1 dot. Total: 8+6=14, +4=18, +2=20, +1=21. Correct.

Median is the 11th term (since 21 is odd, median is (21+1)/2 = 11th). Now, cumulative:

Up to x=1: 8 terms (1–8)
Up to x=2: 8 + 6 = 14 terms (9–14)
11th term is within x=2? Wait, no—wait, 8 (x=1) + 6 (x=2) =14. So 9th to 14th are x=2. 11th is in x=2? But wait, maybe the center (median) is x=2? Wait, no—wait, maybe I made a mistake. Wait, let's list the data:

x=1: 8 times (values: 1,1,1,1,1,1,1,1)

x=2: 6 times (2,2,2,2,2,2)

x=3: 4 times (3,3,3,3)

x=4: 2 times (4,4)

x=5: 1 time (5)

Now, list all 21 values:

1 (8), 2 (6), 3 (4), 4 (2), 5 (1).

Order: [1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,5]

Now, the 11th term: let's index from 1:

1:1, 2:1, 3:1, 4:1, 5:1, 6:1, 7:1, 8:1, 9:2, 10:2, 11:2. Wait, no—wait, 8 ones, then 6 twos. So positions 1–8: 1s, 9–14: 2s. So 11th term is 2? Wait, but that seems off. Wait, maybe the center (median) is 2? Wait, but let's check again. Wait, maybe I miscounted the dots. Let's look at the dot plot:

x=1: how many dots? The vertical column: let's count the dots. The first column (x=1) has 8 dots (since 8 vertical dots). x=2: 6 dots. x=3: 4 dots. x=4: 2 dots. x=5:1 dot. Yes. So the data is: 8 ones, 6 twos, 4 threes, 2 fours, 1 five.

Now, the median is the middle value. For 21 data points, the middle is the 11th value. Let's list the values in order:

1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,5.

Now, count the 11th term:

1 (1st–8th), then 2 (9th–14th). So 11th term is 2. Wait, but that seems correct? Wait, but maybe the center is the median, which is 2? Wait, but let's check the cumulative frequency again. 8 (x=1) + 6 (x=2) =14, which is more than 11, so the 11th term is in x=2. So the median (center) is 2.

Wait, but maybe I made a mistake in counting the dots. Let's re-express the dot plot:

x=1: 8 dots (so 8 data points: 1)

x=2: 6 dots (6 data points: 2)

x=3: 4 dots (4 data points: 3)

x=4: 2 dots (2 data points: 4)

x=5: 1 dot (1 data point: 5)

Total: 8+6+4+2+1=21. Correct.

Median position: (21+1)/2=11th.

Now, cumulative:

After x=1: 8 data points (positions 1–8)
After x=2: 8+6=14 data points (positions 9–14)

11th position is within positions 9–14, which correspond to x=2. Therefore, the median (center) is 2.