Graphical Analysis In Exercises 21–24, you are asked to compare three data sets.


(c) Estimate the sample standard deviations. Then determine how close each of your estimates is by finding the sample standard deviations.


i. ii. iii.

Verified step by step guidance

1. Step 1: Extract the data points from each stem-and-leaf plot. For example, in the first plot, the data points are 9, 15, 18, 23, 23, 27, 32, 35, and 41. Repeat this process for the other two plots.
2. Step 2: Calculate the mean (average) for each data set. Use the formula: $\mu =\frac{\sum x}{n}$, where $x$ represents each data point and $n$ is the total number of data points.
3. Step 3: Compute the deviations from the mean for each data point in each data set. Subtract the mean from each data point to find the deviation.
4. Step 4: Square each deviation and calculate the average of these squared deviations. This is the variance, calculated using the formula: $\sigma ²=\frac{\sum {x-\mu }^2}{n}$.
5. Step 5: Take the square root of the variance to find the sample standard deviation for each data set. Use the formula: $\sigma =\sqrt{\frac{\sum {x-\mu }^2}{n}}$.

Key Concepts

Sample Standard Deviation

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

Estimating Standard Deviation

Estimating the sample standard deviation involves using visual data representations, such as graphs, to gauge the spread of data points. This can be done by observing the range and clustering of values in the data set. While estimates provide a quick insight, they should be verified with actual calculations for accuracy.

Data Set Comparison

Comparing data sets involves analyzing their statistical properties, such as means, medians, and standard deviations, to understand their differences and similarities. This comparison can reveal trends, patterns, or anomalies within the data, aiding in decision-making or hypothesis testing in research.