3. Describing Data Numerically
Standard Deviation
2:46 minutesProblem 8.c.1c
Textbook Question
Lightning Deaths Listed below are the numbers of deaths from lightning strikes in the United States each year for a sequence of recent and consecutive years. Find the values of the indicated statistics.
46 51 44 51 43 32 38 48 45 27 34 29 26 28 23 26 28 40 16 20
c. standard deviation
Verified step by step guidance1
Step 1: Understand the formula for standard deviation. The standard deviation measures the spread of data around the mean. The formula is: \( \sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}} \), where \( \sigma \) is the standard deviation, \( x_i \) are the individual data points, \( \mu \) is the mean of the data, and \( N \) is the number of data points.
Step 2: Calculate the mean (\( \mu \)) of the data set. Add all the data points together and divide by the total number of data points. For this data set: \( 46, 51, 44, 51, 43, 32, 38, 48, 45, 27, 34, 29, 26, 28, 23, 26, 28, 40, 16, 20 \). Use the formula \( \mu = \frac{\sum x_i}{N} \).
Step 3: Compute the squared differences from the mean for each data point. For each \( x_i \), subtract the mean \( \mu \), then square the result: \( (x_i - \mu)^2 \). Perform this calculation for all data points in the set.
Step 4: Find the average of the squared differences. Sum all the squared differences calculated in Step 3, then divide by the total number of data points \( N \). This gives the variance: \( \text{Variance} = \frac{\sum (x_i - \mu)^2}{N} \).
Step 5: Take the square root of the variance to find the standard deviation. Use the formula \( \sigma = \sqrt{\text{Variance}} \). This final step provides the standard deviation of the data set.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Standard Deviation
Standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range. It is calculated as the square root of the variance, which is the average of the squared differences from the mean.
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Mean
The mean, or average, is a measure of central tendency that is calculated by summing all the values in a dataset and dividing by the number of values. It provides a single value that represents the center of the data distribution. Understanding the mean is essential for calculating the standard deviation, as it serves as the reference point from which deviations are measured.
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Variance
Variance is a statistical measure that represents the degree of spread in a set of data points. It is calculated by taking the average of the squared differences between each data point and the mean. Variance is a key component in determining standard deviation, as the standard deviation is simply the square root of the variance, providing insight into the data's variability.
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