Problem 21 - Algebra 2 and Trigonometry

Question

The table shows the number of robberies during a given month in 40 different towns of a state. Find the standard deviation based on this sample \(\begin{array}{|c|c|c|c|c|c|c|c|c|c|}\hline \text { Robberies } & {0} & {1} & {2} & {3} & {4} & {5} & {6} & {7} & {8} & {9} & {10} \\ \hline \text { Frequency } & {1} & {1} & {1} & {2} & {2} & {6} & {10} & {7} & {7} & {2} & {1} \\\ \hline\end{array}\)

Step-by-Step Solution

Answer

The standard deviation is approximately 1.58.

Step 1: List Values with Frequencies

The collected data is grouped in the table where each "Robberies" value is paired with a corresponding "Frequency". List each "Robberies" value repeated as many times as its frequency.\[ \{0, 1, 2, 3, 3, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 9, 9, 10\} \]

Step 2: Calculate the Mean

To get the mean, sum all robbery instances and divide by the total number of data points.\[ \bar{x} = \frac{0 \times 1 + 1 \times 1 + 2 \times 1 + 3 \times 2 + 4 \times 2 + 5 \times 6 + 6 \times 10 + 7 \times 7 + 8 \times 7 + 9 \times 2 + 10 \times 1}{1+1+1+2+2+6+10+7+7+2+1} \]\[ \bar{x} = \frac{240}{40} = 6.0 \]

Step 3: Find Squared Deviations

Subtract the mean from each robbery value, square the results, and list them.\[ (0-6)^2, (1-6)^2, (2-6)^2, (3-6)^2, (3-6)^2, (4-6)^2, (4-6)^2, (5-6)^2, ..., (10-6)^2 \] \[ 36, 25, 16, 9, 9, 4, 4, 1, ..., 16 \]

Step 4: Calculate Sum of Squared Deviations

Multiply each squared deviation by its frequency and sum them all.\[ 36 \times 1 + 25 \times 1 + 16 \times 1 + 9 \times 2 + 4 \times 2 + 1 \times 6 + 0 \times 10 + 1 \times 7 + 4 \times 7 + 9 \times 2 + 16 \times 1 = 98 \]

Step 5: Apply Sample Standard Deviation Formula

Use the formula for sample standard deviation, dividing the sum of squared deviations by one less than the number of data points.\[ s = \sqrt{\frac{98}{40-1}} = \sqrt{2.51} \approx 1.58 \]

Step 6: Report the Standard Deviation

The standard deviation of the given dataset is approximately \(1.58\).

Key Concepts

Mean Calculation

The mean is a vital concept in statistics, often serving as a measure of central tendency. It gives a simple average of the values in a dataset and is calculated by summing all the numbers and then dividing by the total count of values.
In the context of the robbery data, each robbery instance is multiplied by its frequency to account for repetition, and then these products are summed up. This total is then divided by the number of data points, in this case, 40 towns.

• For example: The contribution of 0 robberies (with a frequency of 1) would be calculated as \( 0 \times 1 = 0 \), and similarly for other values.

The calculated mean is \( \bar{x} = 6.0 \), indicating that, on average, each town experienced 6 robberies.

Squared Deviations

Once the mean is calculated, the next step is to assess the spread of the data through squared deviations. Each robbery count is compared to the mean by subtracting the mean from the robbery value and squaring the result. This ensures that the magnitude of differences is considered, removing any directional bias that simple differences might create.

• Squared deviations capture variability; larger squares mean data points are further from the mean.

For example, for 0 robberies, this squared deviation is calculated as \( (0 - 6)^2 = 36 \). These deviations show how individual data points deviate from the average value of 6.

Frequency Distribution

Frequency distribution is a way to represent data to understand its structure better. It underscores how often each data point occurs in a dataset. This can take forms like tables, graphs, or histograms.
The robbery data is provided with frequencies, which tells us how often each number of robberies was recorded across towns.

• For example, 6 robberies occurred in 10 different towns, while 9 robberies occurred twice.

Understanding frequency distribution helps in comprehending how spread out or clustered the data is, offering insight into the commonness of specific values. This is especially useful for calculating means and deviations effectively.

Sample Standard Deviation Formula

The sample standard deviation is a statistic that quantifies the amount of variation or dispersion of a dataset relative to its mean. It uses the formula: \[ s = \sqrt{ \frac{\sum{(x_i - \bar{x})^2}}{n-1} } \] where \( x_i \) represents each individual data point, \( \bar{x} \) is the mean, and \( n \) is the total number of observations.
In this exercise, the sum of squared deviations is divided by \( n - 1 \) to account for the sample being a subset of a larger population. This is known as Bessel's correction, helping provide a more unbiased estimator of the population standard deviation.

• Our problem finds \( s \approx 1.58 \), showing there’s some variability in robberies among towns.

This result allows students to understand the dispersion of robberies, indicating the degree to which robbery occurrences differ from the average.