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

Verified step by step guidance

1. Step 1: Compute the class midpoints for each interval. The class midpoint is calculated as the average of the lower and upper boundaries of each class. For example, for the interval 0–14, the midpoint is (0 + 14) / 2 = 7. Repeat this for all intervals.
2. Step 2: Multiply each class midpoint by its corresponding frequency to compute the product f * x for each class. Then, sum all these products to find Σ(f * x).
3. Step 3: Square each class midpoint to compute x² for each class. Multiply these squared midpoints by their corresponding frequencies to compute f * x² for each class. Then, sum all these products to find Σ(f * x²).
4. Step 4: Calculate the total number of sample values, n, by summing all the frequencies. For example, n = 6 + 18 + 14 + 5 + 5 + 1 + 1.
5. 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 and simplify to compute the standard deviation.

Key Concepts

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 representation helps in understanding the distribution and patterns within the data, making it easier to calculate 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. The midpoints are used in the standard deviation formula to represent the data points within each class, allowing for a more accurate calculation of dispersion in grouped data.