3. Describing Data Numerically

Standard Deviation

3. Describing Data Numerically

Standard Deviation

2:09 minutes

Problem 3.2.41

Textbook Question

The Empirical Rule Based on Data Set 1 “Body Data” in Appendix B, blood platelet counts of women have a bell-shaped distribution with a mean of 255.1 and a standard deviation of 65.4. (All units are 1000 cells/) Using the empirical rule, what is the approximate percentage of women with platelet counts

a. within 2 standard deviations of the mean, or between 124.3 and 385.9?

Verified step by step guidance

Step 1: Recall the Empirical Rule, which states that for a bell-shaped (normal) distribution: approximately 68% of the data falls within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations.

Step 2: Identify the given values from the problem. The mean (μ) is 255.1, and the standard deviation (σ) is 65.4. The range of interest is within 2 standard deviations of the mean, which corresponds to the interval [μ - 2σ, μ + 2σ].

Step 3: Calculate the lower bound of the interval by subtracting 2 standard deviations from the mean: μ - 2σ = 255.1 - 2(65.4).

Step 4: Calculate the upper bound of the interval by adding 2 standard deviations to the mean: μ + 2σ = 255.1 + 2(65.4).

Step 5: Using the Empirical Rule, conclude that approximately 95% of the data falls within 2 standard deviations of the mean. Therefore, the percentage of women with platelet counts between the calculated lower and upper bounds is approximately 95%.

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Key Concepts

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

Empirical Rule

The Empirical Rule, also known as the 68-95-99.7 rule, states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, about 95% falls within two standard deviations, and around 99.7% falls within three standard deviations. This rule helps in understanding the spread of data in a bell-shaped distribution.

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 context, it helps quantify how platelet counts vary among women.

Normal Distribution

A normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its bell-shaped curve, where the mean, median, and mode are all equal. Understanding this concept is crucial for applying the Empirical Rule effectively.

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Textbook Question

Outliers Identify any of the differences found from Exercise 1 that appear to be outliers. For any outliers, how much of an effect do they have on the mean, median, and standard deviation?

Textbook Question

Estimating Standard Deviation with the Range Rule of Thumb. In Exercises 29–32, refer to the data in the indicated exercise. After finding the range of the data, use the range rule of thumb to estimate the value of the standard deviation. Compare the result to the standard deviation computed using all of the data.

Body Temperatures Refer to Data Set 5 “Body Temperatures” in Appendix B and use the body temperatures for 12:00 AM on day 2.

Textbook Question

Finding Standard Deviation from a Frequency Distribution. In Exercises 37–40, refer to the frequency distribution in the given exercise and compute the standard deviation by using the formula below, where x represents the class midpoint, f represents the class frequency, and n represents the total number of sample values. Also, compare the computed standard deviations to these standard deviations obtained by using Formula 3-4 with the original list of data values: (Exercise 37) 18.5 minutes; (Exercise 38) 36.7 minutes; (Exercise 39) 6.9 years; (Exercise 40) 20.4 seconds.

Standard deviation for frequency distribution

Textbook Question

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a. Find the variance of the population {9 cigarettes, 10 cigarettes, 20 cigarettes}.

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Textbook Question

Standard deviation for frequency distribution

Textbook Question

b. After listing the nine different possible samples of two values selected with replacement, find the sample variance (which includes division by ) for each of them; then find the mean of the nine sample variances s2.

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