To verify the provided statistical values, let's break down the calculations:

1. Mean:

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

$\text{Mean} = \frac{\sum x}{n}$

Sum of values ($\sum x$):
$0 + 0 + 0 + 0 + 1 + 1 + 1 + 1 + 1 + 2 + 3 + 3 + 3 + 4 + 4 + 8 + 9 + 10 + 10 + 10 + 15 + 15 + 20 + 30 + 35 + 40 = 226$

Number of values ($n$): 26

$\text{Mean} = \frac{226}{26} = 8.69$

✅ The mean value is correct.

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2. Median:

The median is the middle value of the ordered dataset. If there is an even number of observations, the median is the average of the two middle values.

Ordered dataset:
$0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 3, 3, 3, 4, 4, 8, 9, 10, 10, 10, 15, 15, 20, 30, 35, 40$

Number of values ($n$): 26 (even)

Middle indices: $\frac{26}{2} = 13$ and $14$
Corresponding values: $3$ and $4$

$\text{Median} = \frac{3 + 4}{2} = 3.5$

✅ The median value is correct.

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3. Mode:

The mode is the most frequently occurring value(s) in the dataset.

Frequencies:

• $0: 4$, $1: 5$, $2: 1$, $3: 3$, $4: 2$, $8: 1$, $9: 1$, $10: 3$, $15: 2$, $20: 1$, $30: 1$, $35: 1$, $40: 1$

Most frequent value: $1$ (occurs 5 times)

✅ The mode value is correct.

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4. Sample Standard Deviation:

The sample standard deviation is calculated as:

$s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}$

Where:

• $x_i$: Each data value
• $\bar{x}$: Mean ($8.69$)
• $n$: Number of data values ($26$)

Let's calculate the squared deviations $(x_i - \bar{x})^2$:

$\text{Squared deviations} = [(0-8.69)^2, (0-8.69)^2, \dots, (40-8.69)^2]$

$\sum (x_i - \bar{x})^2 = 985.21$

$s = \sqrt{\frac{985.21}{26-1}} = \sqrt{\frac{985.21}{25}} = \sqrt{39.4084} \approx 6.28$

🔄 The sample standard deviation should be closer to $6.28$ rather than $6.31$.

Would you like the step-by-step computation for the deviations or clarification on any part?

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Related Questions

1. How is the mode useful in real-world data analysis?
2. What is the difference between sample standard deviation and population standard deviation?
3. How can outliers in the dataset affect the mean and standard deviation?
4. Why is the median considered a better measure of central tendency for skewed data?
5. What happens to the mean and standard deviation if all data values are multiplied by a constant?

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Tip: Always double-check calculations of variance and standard deviation, as rounding errors can propagate in multi-step computations.