In Exercises 13 and 14, find the range, mean, variance, and standard deviation of the population data set.

Drunk Driving The number of alcohol-impaired crash fatalities (in thousands) per year from 2010 through 2019 (Source: National Highway Traffic Safety Administration)
10.1 9.9 10.3 10.1 9.9 10.3 11.0 10.9 10.7 10.1

검증된 단계별 안내

Step 1: Organize the data set. The given data set is: 10.1, 9.9, 10.3, 10.1, 9.9, 10.3, 11.0, 10.9, 10.7, 10.1. Ensure all values are listed correctly and count the total number of data points (n = 10).

Step 2: Calculate the range. The range is the difference between the maximum and minimum values in the data set. Identify the maximum value (11.0) and the minimum value (9.9), then compute the range as:

Range = Max - Min

Step 3: Calculate the mean. The mean is the sum of all data points divided by the total number of data points. Use the formula:

Mean = \frac{\sum x}{n}

, where

\sum x

is the sum of all data points and

n

is the number of data points.

Step 4: Calculate the variance. The variance measures the spread of the data points around the mean. Use the formula for population variance:

σ^2 = \frac{\sum (x - Mean)^{2}}{n}

. Subtract the mean from each data point, square the result, sum all squared differences, and divide by the total number of data points.

Step 5: Calculate the standard deviation. The standard deviation is the square root of the variance. Use the formula:

σ = \sqrt{σ^2}

. Take the square root of the variance calculated in Step 4 to find the standard deviation.

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주요 개념

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Descriptive Statistics

Descriptive statistics summarize and describe the main features of a data set. Key measures include the range, which indicates the difference between the maximum and minimum values, and the mean, which is the average of the data points. These statistics provide a quick overview of the data's central tendency and variability.

Variance and Standard Deviation

Variance measures the spread of data points around the mean, indicating how much the values differ from the average. Standard deviation, the square root of variance, provides a more interpretable measure of dispersion in the same units as the data. Both concepts are essential for understanding the variability within a data set.

Population vs. Sample Statistics

Population statistics refer to data that includes every member of a defined group, while sample statistics are derived from a subset of that population. In this context, since the question specifies a population data set, calculations like mean, variance, and standard deviation will reflect the entire group of alcohol-impaired crash fatalities from 2010 to 2019.