Let's go step by step to solve the questions based on the data provided in the image.

1. Finding the Range:

The range is calculated as the difference between the maximum and minimum values of the dataset.

• Maximum value = 11.5
• Minimum value = 9

$\text{Range} = 11.5 - 9 = 2.5 \text{ years}$

2. Finding the Standard Deviation:

To find the standard deviation, we first calculate the mean (average) and then the variance.

• First, let's list all the values:
  $9.5, 11.5, 11, 10, 11, 10.5, 9, 11, 9.5, 10.5, 10, 11, 10, 9.5, 9.5, 10, 9.5, 10, 9, 9.5$

The number of data points $n = 20$.

1. Calculate the mean (average):

$\text{Mean} = \frac{\text{Sum of all values}}{n}$

$\text{Mean} = \frac{(9.5 + 11.5 + 11 + 10 + 11 + 10.5 + 9 + 11 + 9.5 + 10.5 + 10 + 11 + 10 + 9.5 + 9.5 + 10 + 9.5 + 10 + 9 + 9.5)}{20}$

2. Find the variance and then the square root to get the standard deviation.

I'll calculate that next.

3. Five-number summary:

The five-number summary includes:

• Minimum: 9
• Q1 (First quartile): 9.5
• Median (Q2): 10
• Q3 (Third quartile): 10.625 (rounded)
• Maximum: 11.5

Let me calculate the standard deviation now.### Standard Deviation:

• The mean is approximately $10.1$ years.
• The variance is approximately $0.51$.
• The standard deviation is approximately $0.7$ years (rounded to the nearest tenth).

Summary of Results:

1. Range: 2.5 years
2. Standard Deviation: 0.7 years
3. Five-number summary:
   • Minimum: 9
   • Q1: 9.5
   • Median: 10
   • Q3: 10.625
   • Maximum: 11.5

Do you need further details on these calculations, or any other clarifications?

Related Questions:

1. How is the quartile (Q1, Q3) calculated from a dataset?
2. What is the difference between population and sample standard deviation?
3. How does skewness in a dataset affect the range and standard deviation?
4. Can outliers in a dataset significantly impact the five-number summary?
5. What are other measures of data dispersion apart from standard deviation?

Tip:

For large datasets, tools like Excel or statistical software can simplify calculations for range, standard deviation, and quartiles.