Fractional data points plotted on a number line from 0 to 1, showing distribution of values in eighths.

Let’s solve this step by step.

We are given a number line from 0 to 1, marked in eighths:
0, 1/8, 2/8, 3/8, 4/8, 5/8, 6/8, 7/8, 1

Above each mark, there are “x” marks stacked vertically. Each “x” represents one data point at that value.

Our goal is to find the mean (average) of all the data points shown.

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Step 1: Count how many x’s are above each fraction.

- At 0 → 0 x’s
- At 1/8 → 0 x’s
- At 2/8 → 1 x
- At 3/8 → 4 x’s
- At 4/8 → 2 x’s
- At 5/8 → 3 x’s
- At 6/8 → 0 x’s
- At 7/8 → 0 x’s
- At 1 → 3 x’s

Let me double-check the counts from the image description:

Looking again:

- 2/8: one x → correct
- 3/8: four x’s (stacked) → yes
- 4/8: two x’s → yes
- 5/8: three x’s → yes
- 1: three x’s → yes

Total number of data points = 1 + 4 + 2 + 3 + 3 = let’s add:

1 + 4 = 5
5 + 2 = 7
7 + 3 = 10
10 + 3 = 13

So, total data points = 13

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Step 2: Find the sum of all the values.

Each x contributes its position value to the total sum.

So:

- One x at 2/8 → contribution: 2/8
- Four x’s at 3/8 → 4 × 3/8 = 12/8
- Two x’s at 4/8 → 2 × 4/8 = 8/8
- Three x’s at 5/8 → 3 × 5/8 = 15/8
- Three x’s at 1 → 3 × 1 = 3 = 24/8 (since we’ll use eighths for common denominator)

Wait — better to convert everything to eighths to make addition easy.

Note: 1 = 8/8, so 3 × 1 = 3 = 24/8? No! Wait:

Actually, 1 = 8/8, so 3 × 1 = 3 = 24/8? Let's check:

3 = 24/8? → 24 ÷ 8 = 3 → YES.

But actually, since we’re adding fractions, let’s keep it as:

Sum = (1 × 2/8) + (4 × 3/8) + (2 × 4/8) + (3 × 5/8) + (3 × 8/8)

Because 1 = 8/8.

Now compute each term:

1 × 2/8 = 2/8
4 × 3/8 = 12/8
2 × 4/8 = 8/8
3 × 5/8 = 15/8
3 × 8/8 = 24/8

Now add them up:

2 + 12 = 14
14 + 8 = 22
22 + 15 = 37
37 + 24 = 61

So total sum = 61/8

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Step 3: Compute the mean.

Mean = Total Sum ÷ Number of Data Points
= (61/8) ÷ 13
= 61/(8 × 13)
= 61/104

Can this be simplified?

Check if 61 and 104 have any common factors.

61 is a prime number (only divisible by 1 and itself).

104 ÷ 61 ≈ 1.7 → not integer.

So 61/104 is already in simplest form.

But let’s verify our count again — maybe I miscounted the x’s.

Rechecking the original image description:

From left to right:

- 0: no x
- 1/8: no x
- 2/8: one x → ✔️
- 3/8: four x’s (stacked) → ✔️
- 4/8: two x’s → ✔️
- 5/8: three x’s → ✔️
- 6/8: no x
- 7/8: no x
- 1: three x’s → ✔️

Total x’s: 1+4+2+3+3 = 13 → correct.

Sum:

2/8 + 12/8 + 8/8 + 15/8 + 24/8 = (2+12+8+15+24)/8 = 61/8 → correct.

Mean = 61/8 ÷ 13 = 61/(8×13) = 61/104

Is 61/104 reducible? GCD(61,104):

Factors of 61: 1, 61
Factors of 104: 1, 2, 4, 8, 13, 26, 52, 104
No common factor other than 1 → so yes, 61/104 is simplified.

Alternatively, we can write it as a decimal to check reasonableness.

61 ÷ 104 ≈ ?

104 × 0.5 = 52
61 - 52 = 9 → so 0.5 + 9/104 ≈ 0.5 + 0.0865 ≈ 0.5865

Does that make sense? The data points range from 2/8=0.25 to 1, with most around 3/8 to 5/8, and some at 1. So average around 0.58 seems reasonable.

Another way: let’s list all 13 values explicitly to verify.

Values:

One 2/8 → 0.25
Four 3/8 → 0.375 each → 4 × 0.375 = 1.5
Two 4/8 → 0.5 each → 2 × 0.5 = 1.0
Three 5/8 → 0.625 each → 3 × 0.625 = 1.875
Three 1 → 3.0

Now sum: 0.25 + 1.5 = 1.75
1.75 + 1.0 = 2.75
2.75 + 1.875 = 4.625
4.625 + 3.0 = 7.625

Total sum = 7.625

Number of points = 13

Mean = 7.625 ÷ 13

Calculate: 13 × 0.5 = 6.5
7.625 - 6.5 = 1.125
1.125 ÷ 13 = ? 13 × 0.0865 ≈ 1.1245 → so total ≈ 0.5865

Now, 61/104 = ? 61 ÷ 104 = 0.586538... → matches.

Also, 7.625 = 61/8? Let’s see: 61 ÷ 8 = 7.625 → yes!

So 7.625 / 13 = (61/8)/13 = 61/104 → confirmed.

Therefore, the mean is 61/104.

Final Answer:
61/104