Let's start by finding the range, standard deviation, and variance for the given samples: 67, 57, 58, 75, 82, 83, 77, 55, 62.

1. Range

The range is the difference between the highest and lowest values in the data set.

• Highest value: 83
• Lowest value: 55

$\text{Range} = 83 - 55 = 28$

2. Mean

The mean (average) is calculated by summing all the values and dividing by the number of values.

$\text{Mean} = \frac{67 + 57 + 58 + 75 + 82 + 83 + 77 + 55 + 62}{9} = \frac{616}{9} \approx 68.4$

3. Variance

Variance is calculated as the average of the squared differences from the mean.

1. Find the differences from the mean:

   • $67 - 68.4 = -1.4$
   • $57 - 68.4 = -11.4$
   • $58 - 68.4 = -10.4$
   • $75 - 68.4 = 6.6$
   • $82 - 68.4 = 13.6$
   • $83 - 68.4 = 14.6$
   • $77 - 68.4 = 8.6$
   • $55 - 68.4 = -13.4$
   • $62 - 68.4 = -6.4$

2. Square each difference:

   • $(-1.4)^2 = 1.96$
   • $(-11.4)^2 = 129.96$
   • $(-10.4)^2 = 108.16$
   • $(6.6)^2 = 43.56$
   • $(13.6)^2 = 184.96$
   • $(14.6)^2 = 213.16$
   • $(8.6)^2 = 73.96$
   • $(-13.4)^2 = 179.56$
   • $(-6.4)^2 = 40.96$

3. Calculate the mean of these squared differences:

$\text{Variance} = \frac{1.96 + 129.96 + 108.16 + 43.56 + 184.96 + 213.16 + 73.96 + 179.56 + 40.96}{9} \approx \frac{976.24}{9} \approx 108.5$

4. Standard Deviation

The standard deviation is the square root of the variance.

$\text{Standard Deviation} = \sqrt{108.5} \approx 10.4$

Summary

• Range: 28
• Variance: 108.5
• Standard Deviation: 10.4

Would you like more details or have any questions about this calculation? Here are some related questions that might be useful for further exploration:

1. How do you interpret the standard deviation in the context of this data?
2. What are some other measures of central tendency besides the mean?
3. How can you use variance and standard deviation to assess the spread of data?
4. What are some real-world applications of standard deviation?
5. How would these calculations change if an additional value was added to the data set?
6. What is the difference between population and sample variance?
7. How do outliers affect the mean and standard deviation?
8. What are some visual methods to represent the variability of data, such as box plots?

Tip: When calculating variance and standard deviation, it's important to distinguish between sample and population data, as the formulas slightly differ (e.g., using $n-1$ for sample variance instead of $n$ for population variance).