Table of contents

1. Intro to Stats and Collecting Data55m
2. Describing Data with Tables and Graphs1h 55m
3. Describing Data Numerically1h 45m
4. Probability2h 16m
5. Binomial Distribution & Discrete Random Variables2h 33m
6. Normal Distribution and Continuous Random Variables1h 38m
7. Sampling Distributions & Confidence Intervals: Mean1h 53m
8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- Sampling Distribution of Sample Proportion
  29m
- Confidence Intervals for Population Proportion
  42m
9. Hypothesis Testing for One Sample2h 19m
10. Hypothesis Testing for Two Samples3h 22m
11. Correlation1h 6m
12. Regression1h 4m
13. Chi-Square Tests & Goodness of Fit1h 20m
14. ANOVA1h 0m

3. Describing Data Numerically

Standard Deviation

Statistics

3. Describing Data Numerically

Standard Deviation

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3:09 minutes

Problem 6.c.1c

Textbook Question

In Exercises 1 and 2, use the following wait times (minutes) at 10:00 AM for the Tower of Terror ride at Disney World (from Data Set 33 “Disney World Wait Times” in Appendix B).

35 35 20 50 95 75 45 50 30 35 30 30

c. Find the standard deviation s.

Verified step by step guidance

Step 1: Calculate the mean (average) of the data set. Add all the wait times together and divide by the total number of data points. Use the formula:

μ = \sum x

_in, where

n

is the number of data points.

Step 2: Subtract the mean from each data point to find the deviation of each value from the mean. For each data point

x

_i, compute

x

_i - μ.

Step 3: Square each deviation to eliminate negative values. For each deviation, compute

(x

_i - μ)².

Step 4: Find the variance by calculating the average of the squared deviations. Use the formula:

σ

² = ∑(x_i - μ)²n.

Step 5: Take the square root of the variance to find the standard deviation. Use the formula:

s = \sqrt{σ}

².

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Key Concepts

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

Standard Deviation

Standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a set of data 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. It is calculated as the square root of the variance, which is the average of the squared differences from the mean.

Mean

The mean, often referred to as the average, is calculated by summing all the data points and dividing by the number of points. It provides a central value for the data set and is essential for calculating the standard deviation. Understanding the mean helps in interpreting how individual data points relate to the overall dataset.

Variance

Variance is a measure of how much the values in a data set differ from the mean. It is calculated by taking the average of the squared differences between each data point and the mean. Variance is a key component in determining the standard deviation, as it provides insight into the spread of the data, which is crucial for understanding the reliability and consistency of the data set.

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Related Practice

Textbook Question

Exercises 1–10 are based on the following sample data consisting of costs of dinner (dollars) and the amounts of tips (dollars) left by diners. The data were collected by students of the author.

Predictions The sample data result in a linear correlation coefficient of r = 0.846 and the regression equation y^ = -0.00777 + 0.145x. What is the best predicted amount of tip, given that the cost of dinner was $84.62? How was the predicted value found?

Textbook Question

Interpreting the Coefficient of Determination

In Exercises 5–8, use the value of the linear correlation coefficient r to find the coefficient of determination and the percentage of the total variation that can be explained by the linear relationship between the two variables.

Times of Taxi Rides and Fares r = 0.953 (x = time in minutes, y = fare in dollars)

Textbook Question

Interpreting the Coefficient of Determination

Times of Taxi Rides and Tips r = 0.298 (x = time in minutes, y = the amount of tip in dollars)

Textbook Question

Planets The planets of the solar system have the numbers of moons listed below in order from the sun. (Pluto is not included because it was uninvited from the solar system party in 2006.) Include appropriate units whenever relevant.

0 0 1 2 17 28 21 8

e. Find the standard deviation.

f. Find the variance.

Textbook Question

In Exercises 1 and 2, use the following wait times (minutes) at 10:00 AM for the Tower of Terror ride at Disney World (from Data Set 33 “Disney World Wait Times” in Appendix B).

35 35 20 50 95 75 45 50 30 35 30 30

d. Find the variance.

Textbook Question

Lightning Deaths Listed below are the numbers of deaths from lightning strikes in the United States each year for a sequence of recent and consecutive years. Find the values of the indicated statistics.

46 51 44 51 43 32 38 48 45 27 34 29 26 28 23 26 28 40 16 20

c. standard deviation

Textbook Question

46 51 44 51 43 32 38 48 45 27 34 29 26 28 23 26 28 40 16 20

d. Variance

Textbook Question

46 51 44 51 43 32 38 48 45 27 34 29 26 28 23 26 28 40 16 20

e. Range

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