Let's calculate the range, sample standard deviation, and inter-quartile range (IQR) of the given data set.

Given dataLet's calculate the range, sample standard deviation, and inter-quartile range (IQR) of the given data set.

Given

$33, 41, 61, 61, 66, 78, 93, 95, 96, 99$

Step 1: Find the Range

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

• Maximum = 99
• Minimum = 33

$\text{Range} = 99 - 33 = 66$

Step 2: Find the Sample Standard Deviation

The sample standard deviation formula is: $s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2}$ Where:

• $n$ is the sample size
• $x_i$ is each data point
• $\bar{x}$ is the sample mean

First, calculate the mean $\bar{x}$: $\bar{x} = \frac{33 + 41 + 61 + 61 + 66 + 78 + 93 + 95 + 96 + 99}{10} = \frac{723}{10} = 72.3$

Now calculate the squared deviations from the mean for each data point: $\sum (x_i - \bar{x})^2 = (33 - 72.3)^2 + (41 - 72.3)^2 + \dots + (99 - 72.3)^2$

I'll compute the sample standard deviation next.

Step 3: Find the Inter-Quartile Range (IQR)

The IQR is the difference between the first quartile $Q_1$ (25th percentile) and the third quartile $Q_3$ (75th percentile).

I'll calculate the exact values for $Q_1$ and $Q_3$ from the sorted data set and find the IQR.

Let me calculate everything now.### Summary of Calculations:

1. Range: 66
2. Sample Standard Deviation: 23.7 (rounded to one decimal place)
3. Inter-Quartile Range (IQR): 33.5

Would you like further details or explanations on any of the calculations?

Here are 5 related questions to consider:

1. What is the difference between population standard deviation and sample standard deviation?
2. How does the IQR relate to the spread of data?
3. Can the range be affected by outliers more than the IQR?
4. What are the other measures of data spread besides range, IQR, and standard deviation?
5. How do quartiles help in understanding the distribution of data?

Tip: IQR is a robust measure of variability, especially in datasets with outliers, as it focuses on the middle 50% of the data.