To compute the range, variance, and standard deviation for both Group 1 and Group 2, let's use the formulas provided.

1. Range (R)

The range is the difference between the highest value (HV) and the lowest value (LV). $R = HV - LV$

For Group 1:

• Highest value (HV) = 78
• Lowest value (LV) = 72 $R_{Group1} = 78 - 72 = 6$

For Group 2:

• Highest value (HV) = 84
• Lowest value (LV) = 67 $R_{Group2} = 84 - 67 = 17$

2. Variance (s²)

The variance is given by the formula: $s² = \frac{\sum (x - \overline{x})^2}{n-1}$ Where:

• $x$ is each data point
• $\overline{x}$ is the mean of the group
• $n$ is the number of data points

Group 1:

• Mean $\overline{x}_{Group1} = \frac{72+73+76+76+78}{5} = 75$
• Variance calculation: $s²_{Group1} = \frac{(72 - 75)^2 + (73 - 75)^2 + (76 - 75)^2 + (76 - 75)^2 + (78 - 75)^2}{5 - 1}$ $s²_{Group1} = \frac{(-3)^2 + (-2)^2 + (1)^2 + (1)^2 + (3)^2}{4}$ $s²_{Group1} = \frac{9 + 4 + 1 + 1 + 9}{4} = \frac{24}{4} = 6$

Group 2:

• Mean $\overline{x}_{Group2} = \frac{67 + 72 + 76 + 76 + 84}{5} = 75$
• Variance calculation: $s²_{Group2} = \frac{(67 - 75)^2 + (72 - 75)^2 + (76 - 75)^2 + (76 - 75)^2 + (84 - 75)^2}{5 - 1}$ $s²_{Group2} = \frac{(-8)^2 + (-3)^2 + (1)^2 + (1)^2 + (9)^2}{4}$ $s²_{Group2} = \frac{64 + 9 + 1 + 1 + 81}{4} = \frac{156}{4} = 39$

3. Standard Deviation (s)

The standard deviation is the square root of the variance: $s = \sqrt{s²}$

For Group 1: $s_{Group1} = \sqrt{6} \approx 2.45$

For Group 2: $s_{Group2} = \sqrt{39} \approx 6.24$

Summary of Results:

• Group 1:
  • Range = 6
  • Variance = 6
  • Standard Deviation ≈ 2.45
• Group 2:
  • Range = 17
  • Variance = 39
  • Standard Deviation ≈ 6.24

Would you like further details or clarifications?

Related Questions:

1. How is standard deviation used in analyzing data sets?
2. What are some alternative ways to measure data spread other than variance and standard deviation?
3. Why is the range not always the best measure of variability?
4. How do changes in data points affect variance and standard deviation?
5. How would the variance change if we had a larger sample size?

Tip:

Variance and standard deviation are sensitive to outliers, which can greatly affect the overall results. Always check for outliers when interpreting these statistics!