Finding Standard Deviation from a Frequency Distribution. In Exercises 37–40, refer to the frequency distribution in the given exercise and compute the standard deviation by using the formula below, where x represents the class midpoint, f represents the class frequency, and n represents the total number of sample values. Also, compare the computed standard deviations to these standard deviations obtained by using Formula 3-4 with the original list of data values: (Exercise 37) 18.5 minutes; (Exercise 38) 36.7 minutes; (Exercise 39) 6.9 years; (Exercise 40) 20.4 seconds.

Standard deviation for frequency distribution

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Step 1: Calculate the class midpoints for each class interval. The midpoint is found by averaging the lower and upper boundaries of each class interval. For example, for the interval 70–89, the midpoint is (70 + 89) / 2 = 79.5.

Step 2: Multiply each class midpoint by its corresponding frequency to calculate the product f * x for each class. Then, sum all these products to find Σ(f * x).

Step 3: Square each class midpoint to calculate x², then multiply each squared midpoint by its corresponding frequency to calculate f * x² for each class. Sum all these products to find Σ(f * x²).

Step 4: Compute the total number of sample values, n, by summing all the frequencies. For example, n = Σ(f).

Step 5: Use the formula for the standard deviation: s = sqrt((n * Σ(f * x²) - (Σ(f * x))²) / (n * (n - 1))). Substitute the values of n, Σ(f * x), and Σ(f * x²) into the formula to calculate the standard deviation.

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Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. It quantifies how much the values in a dataset deviate from the mean. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates a wider spread of values.

Frequency Distribution

A frequency distribution is a summary of how often each value occurs in a dataset. It organizes data into classes or intervals, showing the number of observations (frequency) within each class. This helps in understanding the distribution of data points and is essential for calculating statistics like the mean and standard deviation.

Class Midpoint

The class midpoint is the value that lies in the middle of a class interval in a frequency distribution. It is calculated by averaging the upper and lower boundaries of the class. Class midpoints are used in the calculation of the standard deviation for grouped data, as they represent the data points within each interval.