Estimating Standard Deviation Both data sets shown in the stem-and-leaf plots have a mean of 165. One has a standard deviation of 16, and the other has a standard deviation of 24. By looking at the stem-and-leaf plots, which is which? Explain your reasoning.

Verified step by step guidance

1. Step 1: Understand the concept of standard deviation. Standard deviation measures the spread of data points around the mean. A smaller standard deviation indicates that the data points are closer to the mean, while a larger standard deviation indicates that the data points are more spread out.
2. Step 2: Analyze the first stem-and-leaf plot. Observe the spread of the data points around the mean (165). Note that the values range from 128 to 207, and the data points appear to be more spread out, with larger gaps between values.
3. Step 3: Analyze the second stem-and-leaf plot. Observe the spread of the data points around the mean (165). Note that the values range from 131 to 192, and the data points appear to be closer to the mean, with smaller gaps between values.
4. Step 4: Compare the spreads of the two plots. The first plot has a wider range and more variability in the data points, suggesting a larger standard deviation. The second plot has a narrower range and less variability, suggesting a smaller standard deviation.
5. Step 5: Conclude which plot corresponds to which standard deviation. Based on the analysis, the first plot likely corresponds to the standard deviation of 24, and the second plot likely corresponds to the standard deviation of 16.

Key Concepts

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range. In this question, the two data sets have different standard deviations, which affects how tightly the data points cluster around the mean.

Stem-and-Leaf Plot

A stem-and-leaf plot is a method of displaying quantitative data in a graphical format, similar to a histogram, to retain the original data values while showing their distribution. Each number is split into a 'stem' (the leading digit or digits) and a 'leaf' (the trailing digit). This visualization helps in comparing distributions and identifying the spread of data, which is crucial for determining the standard deviation.

Mean

The mean, or average, is calculated by summing all the values in a data set and dividing by the number of values. In this question, both data sets have the same mean of 165, which indicates that the central tendency is identical. However, the differing standard deviations suggest that the data points are distributed differently around this mean, which can be inferred from the stem-and-leaf plots.