Let's calculate the required statistics for the given data:

Data:

20, 23, 26, 27, 29, 31, 35, 35

1. Mean:

The mean is the sum of the values divided by the number of values.

$\text{Mean} = \frac{20 + 23 + 26 + 27 + 29 + 31 + 35 + 35}{8}$

$\text{Mean} = \frac{206}{8} = 25.8$

2. Median:

To find the median, the data must be ordered. The ordered data is already given:

20, 23, 26, 27, 29, 31, 35, 35

Since there are 8 numbers (an even number), the median is the average of the two middle numbers (the 4th and 5th values).

$\text{Median} = \frac{27 + 29}{2} = \frac{56}{2} = 28.0$

3. Mode:

The mode is the number that appears most frequently. Here, the number 35 appears twice, while all other numbers appear once.

$\text{Mode} = 35$

4. Standard Deviation:

The standard deviation is a measure of the spread of the numbers. We calculate it using the formula:

$\text{Standard Deviation} = \sqrt{\frac{\sum (x_i - \mu)^2}{n}}$

Where $x_i$ represents each data point, $\mu$ is the mean, and $n$ is the number of data points.

1. Subtract the mean from each data point and square the result:

$(20 - 25.8)^2 = (-5.8)^2 = 33.64$ $(23 - 25.8)^2 = (-2.8)^2 = 7.84$ $(26 - 25.8)^2 = (0.2)^2 = 0.04$ $(27 - 25.8)^2 = (1.2)^2 = 1.44$ $(29 - 25.8)^2 = (3.2)^2 = 10.24$ $(31 - 25.8)^2 = (5.2)^2 = 27.04$ $(35 - 25.8)^2 = (9.2)^2 = 84.64$ $(35 - 25.8)^2 = (9.2)^2 = 84.64$

2. Sum the squared differences:

$33.64 + 7.84 + 0.04 + 1.44 + 10.24 + 27.04 + 84.64 + 84.64 = 249.52$

3. Divide by the number of data points $n = 8$:

$\frac{249.52}{8} = 31.19$

4. Take the square root of the result:

$\text{Standard Deviation} = \sqrt{31.19} \approx 5.6$

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Final Results:

• Mean = 25.8
• Median = 28.0
• Mode = 35
• Standard Deviation ≈ 5.6

Would you like more details on any of the calculations or another question? Here's a tip for you:

Tip:

When calculating standard deviation, always ensure you're using the correct formula. For a sample, you divide by the number of data points $n$, but for a population, you would divide by $n-1$. This is known as "Bessel's correction."