Let's calculate the requested statistical measures for the data set:

$\text{Data set: } 25, 45, 73, 16, 34, 98, 35, 45, 26, 2, 56, 97, 12, 445, 23, 63, 110, 12, 17, 41$

1. Range

The range is the difference between the maximum and minimum values in the data set.

$\text{Range} = \text{Maximum Value} - \text{Minimum Value} = 445 - 2 = 443$

2. Mean

The mean is the sum of all data points divided by the number of data points.

$\text{Mean} = \frac{25 + 45 + 73 + 16 + 34 + 98 + 35 + 45 + 26 + 2 + 56 + 97 + 12 + 445 + 23 + 63 + 110 + 12 + 17 + 41}{20} = \frac{1251}{20} = 62.55$

3. Deviation from the Mean

The deviation from the mean for each data point is calculated by subtracting the mean from the data point. This can be expressed as:

$\text{Deviation} = x_i - \text{Mean}$

Where $x_i$ represents each data point.

4. Mean Absolute Deviation (MAD)

The MAD is the average of the absolute deviations from the mean.

$\text{MAD} = \frac{\sum_{i=1}^{n} |x_i - \text{Mean}|}{n}$

Let's calculate it:

$\text{MAD} = \frac{|25 - 62.55| + |45 - 62.55| + \dots + |41 - 62.55|}{20} = \frac{1156.6}{20} = 57.83$

5. Sample Variance

The sample variance is the average of the squared deviations from the mean, divided by $n-1$, where $n$ is the number of data points.

$\text{Sample Variance} = \frac{\sum_{i=1}^{n} (x_i - \text{Mean})^2}{n-1}$

$\text{Sample Variance} = \frac{57370.45}{19} \approx 3019.5$

6. Standard Deviation

The standard deviation is the square root of the variance.

$\text{Standard Deviation} = \sqrt{\text{Sample Variance}} = \sqrt{3019.5} \approx 54.95$

Summary:

• Range: 443
• Mean: 62.55
• Mean Absolute Deviation: 57.83
• Sample Variance: 3019.5
• Standard Deviation: 54.95

Would you like further details or have any questions?

Relative Questions:

1. How is the range affected if a new value, say 500, is added to the data set?
2. What does the mean absolute deviation tell us about the data?
3. How is sample variance different from population variance?
4. Why is the standard deviation a more useful measure than variance in some cases?
5. How would the mean change if the smallest value in the data set were removed?

Tip:

Always remember to check whether you need to calculate sample variance (dividing by $n-1$) or population variance (dividing by $n$) based on the context of your data.