Finding the Sample Mean and Standard Deviation for Grouped Data In Exercises 39 and 40, make a frequency distribution for the data. Then use the table to find the sample mean and the sample standard deviation of the data set.


3 3 5 3 8 0 3 9 6 6 7 1 6 3 2 6 9 1 8 5 0 2 3 4 9
5 8 1 9 7 6 9 6 7 0 6 3 8 6 8 7 3 8 9 3 7 2 4 4 1

Verified step by step guidance

1. Step 1: Create a frequency distribution table. Group the data into intervals (e.g., 0-2, 3-5, 6-8, etc.) and count the frequency of data points in each interval. This will help organize the data for further calculations.
2. Step 2: Calculate the midpoint for each interval. The midpoint is the average of the lower and upper bounds of each interval. For example, for the interval 0-2, the midpoint is (0 + 2) / 2 = 1.
3. Step 3: Compute the product of each midpoint and its corresponding frequency. Multiply the midpoint of each interval by the frequency of that interval. This will be used to calculate the sample mean.
4. Step 4: Calculate the sample mean using the formula: \( \bar{x} = \frac{\sum (f \cdot x)}{\sum f} \), where \( f \) is the frequency and \( x \) is the midpoint. Sum up the products from Step 3 and divide by the total frequency.
5. Step 5: Calculate the sample standard deviation using the formula: \( s = \sqrt{\frac{\sum f (x - \bar{x})^2}{n - 1}} \), where \( \bar{x} \) is the sample mean, \( x \) is the midpoint, \( f \) is the frequency, and \( n \) is the total number of data points. First, compute \( (x - \bar{x})^2 \) for each interval, multiply by the frequency, sum these values, and divide by \( n - 1 \). Finally, take the square root of the result.

Key Concepts

Sample Mean

The sample mean is the average of a set of values, calculated by summing all the data points and dividing by the number of observations. For grouped data, the mean is found by multiplying the midpoint of each class interval by its frequency, summing these products, and then dividing by the total number of observations. This provides a measure of central tendency that represents the data set.

Sample Standard Deviation

The sample standard deviation quantifies the amount of variation or dispersion in a set of values. It is calculated by taking the square root of the variance, which is the average of the squared differences from the mean. For grouped data, the standard deviation involves finding the squared differences between each class midpoint and the sample mean, weighted by the frequency of each class, to assess how spread out the data points are.

Frequency Distribution

A frequency distribution is a summary of how often each value or range of values occurs in a data set. It organizes data into classes or intervals and shows the number of observations (frequency) within each class. This representation helps in visualizing the distribution of data, making it easier to calculate measures like the mean and standard deviation, especially for large data sets.