Ch. 2 - Descriptive Statistics

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 2 - Descriptive Statistics

Problem 2.4.51b

Chapter 2, Problem 2.4.51b

Extending Concepts

Alternative Formula You used SSₓ = Σ(x − x̄)² when calculating variance and standard deviation. An alternative formula that is sometimes more convenient for hand calculations is
SSₓ = Σ x² − (Σ x)² / n.
You can find the sample variance by dividing the sum of squares by n − 1 and the sample standard deviation by finding the square root of the sample variance.

b. Use the alternative formula to calculate the sample standard deviation for the data set in Exercise 15.

Verified step by step guidance

Step 1: Understand the alternative formula for the sum of squares (SSₓ). The formula is SSₓ = Σx² − (Σx)² / n, where Σx² is the sum of the squares of the data values, Σx is the sum of the data values, and n is the number of data points.

Step 2: Calculate Σx² by squaring each data value in the data set and then summing these squared values.

Step 3: Calculate Σx by summing all the data values in the data set. Then, square this sum and divide it by n to compute (Σx)² / n.

Step 4: Subtract (Σx)² / n from Σx² to find the sum of squares (SSₓ).

Step 5: Divide SSₓ by n − 1 to calculate the sample variance. Finally, take the square root of the sample variance to find the sample standard deviation.

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Key Concepts

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

Sum of Squares (SS)

The Sum of Squares (SS) is a measure used in statistics to quantify the total variation within a dataset. It is calculated by summing the squared differences between each data point and the mean of the dataset. This value is crucial for determining variance and standard deviation, as it reflects how much the data points deviate from the average.

Sample Variance

Sample variance is a statistic that measures the dispersion of a sample dataset. It is calculated by dividing the Sum of Squares (SS) by the number of observations minus one (n - 1), which corrects for bias in the estimation of the population variance. This adjustment is important because it provides a more accurate representation of variability when working with a sample rather than the entire population.

Standard Deviation

Standard deviation is a widely used measure of the amount of variation or dispersion in a set of values. It is the square root of the variance, providing a measure that is in the same units as the original data. A low standard deviation indicates that the data points tend to be close to the mean, while a high standard deviation indicates greater spread among the values.

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Related Practice

Textbook Question

Protein Powder During a quality assurance check, the actual contents (in grams) of six containers of protein powder were recorded as 1525, 1526, 1502, 1516, 1529, and 1511.

a. Find the mean and the median of the contents.

100

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Textbook Question

Pearson’s Index of Skewness The English statistician Karl Pearson (1857–1936) introduced a formula for the skewness of a distribution.

P = 3 (x̄ - median) / s

Most distributions have an index of skewness between -3 and 3. When P > 0, the data are skewed right. When P < 0, the data are skewed left. When P = 0, the data are symmetric. Calculate the coefficient of skewness for each distribution. Describe the shape of each.

a. x̄ = 17, s = 2.3, median = 19

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Textbook Question

Shifting Data Sample annual salaries (in thousands of dollars) for employees at a company are listed.

40 35 49 53 38 39 40

37 49 34 38 43 47 35

a. Find the sample mean and the sample standard deviation.

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Textbook Question

Life Spans of Fruit Flies The life spans of a species of fruit fly have a bell-shaped distribution, with a mean of 33 days and a standard deviation of 4 days.

b. The life spans of three randomly selected fruit flies are 29 days, 41 days, and 25 days. Using the Empirical Rule, find the percentile that corresponds to each life span.

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Textbook Question

Scaling Data Sample annual salaries (in thousands of dollars) for employees at a company are listed.

42 36 48 51 39 39 42

36 48 33 39 42 45 50

b. Each employee in the sample receives a 5% raise. Find the sample mean and the sample standard deviation for the revised data set.

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Textbook Question

Drawing a Box-and-Whisker Plot In Exercises 15–18,

(b) draw a box-and-whisker plot that represents the data set.

2 7 1 3 1 2 8 9 9 2 5 4 7 3 7 5 4

2 3 5 9 5 6 3 9 3 4 9 8 8 2 3 9 5

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