Use frequency distribution formulas to estimate the sample mean and the sample standard deviation of the data set in Exercise 2.

Verified step by step guidance

1. Step 1: Organize the data into a frequency distribution table. This table should include columns for class intervals, frequencies (f), midpoints of the class intervals (x), and any other necessary values for calculations.
2. Step 2: Calculate the midpoint (x) for each class interval. The midpoint is the average of the lower and upper boundaries of the class interval, calculated as \( x = \frac{{\text{lower boundary} + \text{upper boundary}}}{2} \).
3. Step 3: Compute the weighted sum of the midpoints by multiplying each midpoint (x) by its corresponding frequency (f). Sum these products to find \( \sum f \cdot x \).
4. Step 4: Estimate the sample mean using the formula \( \text{Sample Mean} = \frac{{\sum f \cdot x}}{\sum f} \), where \( \sum f \) is the total frequency.
5. Step 5: Estimate the sample standard deviation using the formula \( \text{Sample Standard Deviation} = \sqrt{\frac{{\sum f \cdot (x - \text{Sample Mean})^2}}{\sum f}} \). For this, calculate \( (x - \text{Sample Mean})^2 \) for each class interval, multiply by the frequency (f), sum these values, and divide by \( \sum f \). Finally, take the square root of the result.

Key Concepts

Frequency Distribution

A frequency distribution is a summary of how often each value occurs in a dataset. It organizes data into categories or intervals, allowing for easier analysis of patterns and trends. Understanding frequency distributions is essential for calculating measures like the sample mean and standard deviation, as it provides a clear view of data distribution.

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. It serves as a measure of central tendency, providing insight into the overall level of the data. In the context of frequency distributions, the sample mean can be estimated by weighting the midpoints of intervals by their frequencies.

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 measures how far each data point is from the sample mean. In frequency distributions, the standard deviation can be estimated using the frequencies and midpoints of the intervals, reflecting the spread of the data.