Let's calculate the mean, variance, and standard deviation for the given data set:

Data Set:

$2, 5, 4, 3, 3, 5, 3, 5, 3, 3$

1. Mean (Average):

The mean is calculated by summing all the data points and dividing by the number of data points.

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

Where:

• $\sum x_i$ is the sum of all data points.
• $n$ is the number of data points.

Let's sum the numbers: $2 + 5 + 4 + 3 + 3 + 5 + 3 + 5 + 3 + 3 = 38$

There are 10 data points.

$\text{Mean} = \frac{38}{10} = 3.8$

So, the mean is 3.8.

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2. Variance:

Variance is a measure of how much the data points deviate from the mean. To calculate the variance, use the formula:

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

Where:

• $\mu$ is the mean.
• $x_i$ are the data points.
• $n$ is the number of data points.

First, let's calculate the squared differences from the mean:

$(2 - 3.8)^2 = (-1.8)^2 = 3.24$ $(5 - 3.8)^2 = (1.2)^2 = 1.44$ $(4 - 3.8)^2 = (0.2)^2 = 0.04$ $(3 - 3.8)^2 = (-0.8)^2 = 0.64$ $(3 - 3.8)^2 = (-0.8)^2 = 0.64$ $(5 - 3.8)^2 = (1.2)^2 = 1.44$ $(3 - 3.8)^2 = (-0.8)^2 = 0.64$ $(5 - 3.8)^2 = (1.2)^2 = 1.44$ $(3 - 3.8)^2 = (-0.8)^2 = 0.64$ $(3 - 3.8)^2 = (-0.8)^2 = 0.64$

Now, sum these squared differences: $3.24 + 1.44 + 0.04 + 0.64 + 0.64 + 1.44 + 0.64 + 1.44 + 0.64 + 0.64 = 10.80$

Finally, calculate the variance: $\text{Variance} = \frac{10.80}{10} = 1.08$

So, the variance is 1.08.

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3. Standard Deviation:

The standard deviation is simply the square root of the variance.

$\text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{1.08} \approx 1.04$

So, the standard deviation is approximately 1.04.

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Summary of Results:

• Mean: 3.8
• Variance: 1.08
• Standard Deviation: 1.04

Would you like more details or have any questions on these calculations?

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Here are some related questions:

1. How would the mean change if a new data point were added to this set?
2. What is the relationship between variance and standard deviation?
3. How do you interpret the standard deviation in the context of this data set?
4. What is the effect of outliers on variance and standard deviation?
5. How can you calculate these statistics using a calculator or software?

Tip: The standard deviation is a better measure of spread than variance because it is in the same units as the data, making it more interpretable.