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Chapter 3: Problem 53

Consider the sample data in the following frequency distribution. $$\begin{array}{ccc} \text { Class } & \text { Midpoint } & \text { Frequency } \\ 3-7 & 5 & 4 \\ 8-12 & 10 & 7 \\ 13-17 & 15 & 9 \\ 18-22 & 20 & 5 \end{array}$$ a. Compute the sample mean. b. Compute the sample variance and sample standard deviation.

Short Answer

Expert verified
Sample mean: 13, Sample variance: 25, Sample standard deviation: 5.

Step by step solution

01

Calculate Total Frequency

To compute the sample mean, we first need the total frequency (\(n\)), which is the sum of all individual frequencies. Adding the frequencies, \(4 + 7 + 9 + 5 = 25\). So, \(n = 25\).
02

Calculate the Weighted Midpoint Sum

To find the sample mean, use the sum of each class鈥檚 midpoint multiplied by its frequency. Calculate as:\(5 \times 4 + 10 \times 7 + 15 \times 9 + 20 \times 5\).Thus, \(20 + 70 + 135 + 100 = 325\).
03

Compute the Sample Mean

The sample mean \(\bar{x}\) is the total weighted sum from Step 2 divided by the total frequency from Step 1. So, \(\bar{x} = \frac{325}{25} = 13\).
04

Calculate Each Midpoint's Squared Deviation

Find the squared deviation of each midpoint from the mean and multiply by its frequency. Calculate as:- For 5: \((5-13)^2 \times 4 = 256\)- For 10: \((10-13)^2 \times 7 = 63\)- For 15: \((15-13)^2 \times 9 = 36\)- For 20: \((20-13)^2 \times 5 = 245\)
05

Compute the Sum of Squared Deviations

Add the results from Step 4 to find the sum of squared deviations: \(256 + 63 + 36 + 245 = 600\).
06

Calculate the Sample Variance

The sample variance \(s^2\) is the sum of squared deviations divided by \(n-1\). Thus, \(s^2 = \frac{600}{24} = 25\).
07

Calculate the Sample Standard Deviation

The sample standard deviation \(s\) is the square root of the variance: \(s = \sqrt{25} = 5\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sample Mean
The sample mean is a measure of the central tendency, which represents the average value of a dataset. Calculating the sample mean with grouped frequency data requires some additional steps compared to simply averaging a set of numbers. Here's a simple way to break it down:
  • Total Frequency: First, identify the total number of observations. In the given frequency distribution, we add up all the frequencies: 4 + 7 + 9 + 5 = 25. This total is key for our calculations since it represents the number of observations in the sample.
  • Weighted Midpoint Sum: For each class, calculate the product of its midpoint and its frequency. The midpoint is the center of each class interval: for instance, 5 is the midpoint for the class 3-7. Multiply each midpoint by its respective frequency, and sum these products: (5 x 4) + (10 x 7) + (15 x 9) + (20 x 5) = 325.
  • Sample Mean: Finally, divide the weighted sum by the total frequency to find the sample mean: 325/25 = 13.
So, the sample mean tells us that the central point of our data is around 13. This number helps us understand where most of our data points are likely clustering around.
Sample Variance
Sample variance measures the spread or variability of the data points relative to the sample mean. A larger variance indicates that the data points tend to be more spread out from the mean, while a smaller variance suggests they are more tightly clustered.
To calculate the sample variance with frequency distribution data, follow these steps:
  • Squared Deviations: First, determine how far each midpoint deviates from the sample mean. For example, for the midpoint 5, the deviation is (5 - 13). Square this deviation to eliminate negative values, giving us squared deviations such as (5 - 13)虏.
  • Weighted Squared Deviations: Now, multiply each squared deviation by its corresponding frequency to weight the results appropriately: (5 - 13)虏 x 4, (10 - 13)虏 x 7, and so on. This step reflects the influence of each class based on how many data points it represents.
  • Sum of Squared Deviations: Add all these weighted squared deviations together to get the total sum: 256 + 63 + 36 + 245 = 600.
  • Calculate Sample Variance: Finally, divide this sum by (n - 1), where n is the total frequency: 600/24 = 25.
This result, 25, tells us how much variation there is in the data set. Understanding variance gives us a picture of how different our values are from each other.
Standard Deviation
The standard deviation is a direct measure of how spread out the numbers are in your dataset. It's derived from the sample variance and is often more intuitive to interpret. Standard deviation gives us a concrete sense of the typical distance between each data point and the mean.
Here's a step-by-step way to understand it:
  • Derived from Variance: Standard deviation is simply the square root of the variance. This transformation helps bring the units of the standard deviation back to the same units as the data, making it more suitable for comparison.
  • Practical Interpretation: In this exercise, after calculating that our sample variance is 25, we take the square root to find the standard deviation: \(s = \sqrt{25} = 5\). This result tells us the average deviation of each data point from the mean is about 5 units.
  • Why It Matters: Knowing the standard deviation aids in understanding the overall variability of the dataset. If most data points fall within one standard deviation of the mean, the dataset is considered less variable and more predictable.
In summary, the standard deviation provides a clear picture of data spread, making it an essential statistic for data analysis.

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Most popular questions from this chapter

A panel of economists provided forecasts of the U.S. economy for the first six months of 2007 (The Wall Street Journal, January 2,2007 ). The percentage changes in gross domestic product (GDP) forecasted by 30 economists are as follows. \\[\begin{array}{lllllllll} 2.6 & 3.1 & 2.3 & 2.7 & 3.4 & 0.9 & 2.6 & 2.8 & 2.0 & 2.4 \\ 2.7 & 2.7 & 2.7 & 2.9 & 3.1 & 2.8 & 1.7 & 2.3 & 2.8 & 3.5 \\ 0.4 & 2.5 & 2.2 & 1.9 & 1.8 & 1.1 & 2.0 & 2.1 & 2.5 & 0.5 \end{array}\\] a. What is the minimum forecast for the percentage change in GDP? What is the maximum? b. Compute the mean, median, and mode. c. Compute the first and third quartiles. d. Did the economists provide an optimistic or pessimistic outlook for the U.S. economy? Discuss.

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