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The sample mean is a fundamental concept in statistics, representing the average value of a sample. It's particularly useful when you need to summarize a large set of data with a single value. In our exercise, we are interested in the mean age of adult shoplifters. To find the sample mean of any set of values, you sum up all the values and divide by the total number of observations.

For group data like age ranges, we use midpoints to estimate the mean. Here, the class midpoints for age ranges provide an estimate of the typical age in each category. For instance, the midpoint for the range 21-30 is 25.5, calculated as \(\frac{21+30}{2}\). Similarly, we compute midpoints for other age ranges and use them in the weighted mean formula:

• Multiply each midpoint by its corresponding frequency (number of individuals).
• Add the results to get a total sum.
• Divide by the total frequency (the sum of all frequencies), which in this case is 895.

Hence, the sample mean age is approximately 35.82 years, meaning on average, adult shoplifters in this sample tend to be around 36 years old.

Sample variance measures how much the ages of shoplifters vary around the sample mean. It's an essential statistic because it helps us understand the spread or dispersion of data in a dataset.

To calculate sample variance, follow these steps:

• First, find the difference between each class midpoint and the calculated sample mean (35.82 years in our case).
• Square these differences to eliminate negative values and give more weight to outliers.
• Multiply each squared difference by the frequency of its corresponding class.
• Sum over all these values to get the "Sum of Squares of Deviations" from the mean.
• Finally, divide by \(n-1\) to account for degrees of freedom (n is the total number of observations).

The calculations yield a sample variance of approximately 60.82. This value suggests how much the age of shoplifters in this sample differs from the mean age.

Age distribution refers to how ages are spread across different categories in a dataset. Understanding age distribution helps identify patterns, trends and potential causes or effects associated with different age groups in studies or reports.

In this exercise, age distribution covers three age ranges: 21-30, 31-40, and 41 and over. By looking at the given data:

• 260 shoplifters are in the 21-30 range,
• 348 in the 31-40 range,
• and 287 are 41 and over.

This spread shows which age groups might account for a higher number of shoplifting incidents. It also informs us about the age group with the most frequent occurrences, which here is the 31-40 age group. Exploring age distributions like this can guide policies, prevention strategies, and further studies.

Standard deviation is a crucial statistical tool that quantifies the amount of variance or dispersion in a set of data points. It tells us how much the individual data points differ from the mean of the dataset.

The standard deviation is derived from the variance by taking the square root of the sample variance. This step brings the measure back to the original unit of measure (in this case, years), making it more directly interpretable.

For the shoplifting age data, with a sample variance of 60.82, the standard deviation is approximately \(s = \sqrt{60.82} \approx 7.80\).

This indicates that the ages of the shoplifters typically vary by about 7.8 years from the average age of 35.82. In simpler terms, most shoplifters' ages are within 7.8 years above or below the mean age, providing a clear snapshot of age variability in this sample. Standard deviation helps paint a clear picture of how concentrated or spread the age data is around the mean.