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: Observe the three histograms provided. Each histogram represents a different data set. The spread and concentration of the data entries will help estimate the sample standard deviation. Wider spreads typically indicate higher standard deviations, while narrower spreads suggest lower standard deviations.
2. Step 2: For the first histogram (i), the data entries are spread across the range 4 to 10, with a peak frequency at 7. The distribution appears moderately spread out, suggesting a medium standard deviation. To calculate the sample standard deviation, use the formula: \( s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \), where \( x_i \) are the data points, \( \bar{x} \) is the mean, and \( n \) is the sample size.
3. Step 3: For the second histogram (ii), the data entries are concentrated at the extremes (4 and 10) with minimal frequencies in the middle. This indicates a larger spread and likely a higher standard deviation compared to the first histogram. Apply the same formula for standard deviation, ensuring to account for the extreme values.
4. Step 4: For the third histogram (iii), the data entries are tightly concentrated around 6, 7, and 8, with no entries outside this range. This suggests a very narrow spread and a low standard deviation. Use the standard deviation formula, noting the limited range of values.
5. Step 5: After estimating the standard deviations for each histogram, calculate the exact values using the formula provided. Compare the calculated values to your initial estimates to determine how close your estimates were to the actual standard deviations.

Key Concepts

Sample Standard Deviation

The sample standard deviation is a measure of the amount of variation or dispersion in a set of sample data points. 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.

Frequency Distribution

A frequency distribution is a summary of how often each value occurs in a dataset. It is often represented graphically using histograms, where the x-axis represents the data entries and the y-axis represents the frequency of those entries. This visual representation helps in understanding the distribution and central tendencies of the data.

Estimation Techniques

Estimation techniques involve using sample data to infer characteristics about a larger population. In the context of standard deviation, one might estimate the sample standard deviation based on visual inspection of the data distribution before calculating it precisely. This approach helps in making quick assessments and comparisons between different datasets.