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Ch. 2 - Descriptive Statistics
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 2 - Descriptive StatisticsProblem 2.4.21c
Chapter 2, Problem 2.4.21c

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. Scatter plot showing data points distributed around the values 10 to 14, indicating variability in the data sets.
ii. A scatter plot showing data points distributed around the values 10 to 14, with a peak at 12.
iii. A dot plot showing data points clustered around the value 12, with values ranging from 10 to 14 on the horizontal axis.

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Step 1: Observe the three graphs provided. Each graph represents a data set with values distributed along the x-axis (10, 11, 12, 13, 14). The y-axis indicates the frequency of each value. The spread of the data points will help estimate the sample standard deviation for each data set.
Step 2: Calculate the mean (average) for each data set. To do this, sum all the values in the data set, weighted by their frequency, and divide by the total number of data points. Use the formula: μ=∑inxiN, where N is the total number of data points.
Step 3: Estimate the sample standard deviation for each data set. The standard deviation measures the spread of the data around the mean. Use the formula: ∑in(xi-μ)2(N-1), where μ is the mean and N is the total number of data points.
Step 4: Compare the spread of the data points visually. For example, in the first graph, the data points are concentrated closer to the mean, suggesting a smaller standard deviation. In the second graph, the data points are more spread out, indicating a larger standard deviation. The third graph appears to have an intermediate spread.
Step 5: After calculating the sample standard deviations for each data set using the formula, compare the calculated values to your visual estimates. This will help verify the accuracy of your estimates and provide insight into the relationship between the spread of data and the standard deviation.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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Estimating Standard Deviation

Estimating the sample standard deviation involves using visual data representations, such as scatter plots or dot plots, to gauge the spread of data points. This estimation can be done by observing the range of values and their distribution, allowing for a rough calculation before computing the exact standard deviation using statistical formulas.
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Data Distribution

Data distribution refers to how values are spread or arranged across a range. In the context of the provided scatter plots, understanding the distribution helps identify patterns, such as clustering around certain values or the presence of outliers. This insight is crucial for interpreting the variability and central tendency of the data sets.
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Related Practice
Textbook Question

Use the data set and the indicated number of classes to construct

(c) a frequency polygon,


Hospitals

Number of classes: 8

Data set: Number of hospitals in each of the 50 U.S. states and 5 inhabited territories (Source: American Hospital Directory) 10 90 51 1 77 341 56 34 8 214 111 3 14 40 18 142 102 55 75 108 72 53 19 105 55 83 1 69 19 108 10 27 14 78 37 31 186 146 90 37 177 52 11 67 25 100 361 35 91 2 7 61 78 33 14

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Textbook Question

Use the ogive to approximate the

the number of black bears that weigh between 158.5 pounds and 244.5 pounds.

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Textbook Question

Use the relative frequency histogram to describe any patterns with the data.

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Textbook Question

What Would You Do? The admissions department for a college is asked to recommend the minimum SAT scores that the college will accept for full-time students. The SAT scores of 50 applicants are listed. 1170 1000 910 870 1070 1290 920 1470 1080 1180 770 900 1120 1070 1370 1160 970 930 1240 1270 1250 1330 1010 1010 1410 1130 1210 1240 960 820 650 1010 1190 1500 1400 1270 1310 1050 950 1150 1450 1290 1310 1100 1330 1410 840 1040 1090 1080

If you want to accept the top 88% of the applicants, what should the minimum score be? Explain.

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Textbook Question

Use the data set and the indicated number of classes to construct


(c) a frequency polygon,

Pulse Rates

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Textbook Question

What Would You Do? You work at a bank and are asked to recommend the amount of cash to put in an ATM each day. You do not want to put in too much (which would cause security concerns) or too little (which may create customer irritation). The daily withdrawals (in hundreds of dollars) for 30 days are listed. 72 84 61 76 104 76 86 92 80 88 98 76 97 82 84 67 70 81 82 89 74 73 86 81 85 78 82 80 91 83

If you are willing to run out of cash on 10% of the days, how much cash should you put in the ATM each day? Explain.

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