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Chapter 8: Problem 1

Compute the (sample) variance and standard deviation of the data samples given in Exercises \(1-8 .\) You calculated the means in the last exercise set. Round all answers to two decimal nlaces. $$ -1,5,5,7,14 $$

Short Answer

Expert verified
The sample variance of the given data set is 29 and the sample standard deviation is approximately 5.39.

Step by step solution

01

(Step 1: Recall formulas for variance and standard deviation)

The formulas for calculating the sample variance and standard deviation are: Sample Variance (s^2) = \(\frac{\Sigma(x_i - \bar{x})^2}{n-1}\) Sample Standard Deviation (s) = \(\sqrt{\text{Sample Variance}}\) where \(\bar{x}\) is the sample mean, \(x_i\) are the individual data points in the sample, and n is the number of data points in the sample.
02

(Step 2: Write down the given data and mean value)

The given data is: $$-1, 5, 5, 7, 14$$ From the previous exercise set, the mean of this data (\(\bar{x}\)) is 6.
03

(Step 3: Apply the formula for calculating the sample variance)

Using the formula for sample variance, substitute the values and compute: Sample Variance (s^2) = \(\frac{\Sigma(x_i - \bar{x})^2}{n-1}\) Sample Variance (s^2) = \(\frac{(-1 - 6)^2 + (5 - 6)^2 + (5 - 6)^2 + (7 - 6)^2 + (14 - 6)^2}{5-1}\) Sample Variance (s^2) = \(\frac{49 + 1 + 1 + 1 + 64}{4}\) Sample Variance (s^2) = \(\frac{116}{4}\) Sample Variance (s^2) = 29
04

(Step 4: Apply the formula for calculating the sample standard deviation)

Using the formula for sample standard deviation, substitute the value of sample variance: Sample Standard Deviation (s) = \(\sqrt{\text{Sample Variance}}\) Sample Standard Deviation (s) = \(\sqrt{29}\) Sample Standard Deviation (s) 鈮� 5.39 (rounded to two decimal places) The sample variance of the given data set is 29 and the sample standard deviation is approximately 5.39.

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Key Concepts

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

Sample Variance
Sample variance is a fundamental concept in statistics that measures how much sample data points deviate from their sample mean. It tells you how spread out the data is, which is essential for understanding variability within your data set.

To calculate the sample variance, follow these steps:
  • First, identify each data point in your set. For example, with the set from our exercise, these points are \(-1, 5, 5, 7, 14\).
  • Next, calculate the mean (average) of these data points. The mean in our example is 6.
  • Subtract the mean from each data point to find the deviation of each point from the mean.
  • Square each of these deviations. Squaring gets rid of negative numbers and emphasizes larger deviations.
  • Sum all the squared deviations.
  • Finally, divide the total by \(n-1\), where \(n\) is the number of data points. This step adjusts for bias in estimating population parameters from a sample, providing what's called the "unbiased sample variance."

After performing these steps, you get the sample variance, a single numeric value that represents the overall diversity of your data set.
Standard Deviation
Standard deviation takes the concept of variance a step further. Instead of using squared units, it translates variability back into the original units of the data, making it easier to interpret in a practical context. It shows how much data tends to deviate, on average, from the mean.

Here's how to calculate the standard deviation based on sample variance:
  • After calculating the sample variance, simply take the square root of this value.
  • The resulting value is the sample standard deviation.
  • For our exercise, the sample standard deviation is approximately 5.39.

This statistic is crucial because it provides transparency into data dispersion, helping in assessments, such as determining if a data point is unusually distant from the mean compared to others. Knowing the standard deviation aids in comparative analysis and can impact decision-making processes effectively.
Data Analysis
Data analysis encompasses statistical techniques for evaluating data to discover insights and support decision-making. The calculation of measures like sample variance and standard deviation plays a vital role in this field.

In practical terms, understanding variance and standard deviation helps us:
  • Interpret how data is distributed in relation to the mean.
  • Identify patterns, anomalies, or trends in data sets.
  • Evaluate consistency or variability in business metrics or experimental results.
  • Make informed predictions about future observations based on historical data.

Accurate data analysis allows researchers and businesses to derive significant trends and make data-driven decisions. Analyzing data using standard deviation and variance helps highlight the smallest to most noticeable fluctuations, giving a comprehensive view of past and present data, which is instrumental in forecasting and strategic planning.

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Most popular questions from this chapter

Your pet tarantula, Spider, has a .12 probability of biting an acquaintance who comes into contact with him. Next week, you will be entertaining 20 friends (all of whom will come into contact with Spider). a. How many guests should you expect Spider to bite? b. At your last party, Spider bit 6 of your guests. Assuming that Spider bit the expected number of guests, how many guests did you have?

Your grade in a recent midterm was \(80 \%\), but the class median was \(100 \%\). Your score was lower than the average score, right?

Calculate the expected value, the variance, and the standard deviation of the given random variable \(X .\) You calculated the expected values in the last exercise set. Round all answers to two decimal places.) Thirty darts are thrown at a dartboard. The probability of hitting a bull's-eye is \(\frac{1}{5}\). Let \(X\) be the number of bull's-eyes hit.

A roulette wheel has the numbers 1 through 36,0 , and \(00 .\) Half of the numbers from 1 through 36 are red, and a bet on red pays even money (that is, if you win, you will get back your \(\$ 1\) plus another \(\$ 1\) ). How much do you expect to win with a \(\$ 1\) bet on red? HINT [See Example 4.]

Pastimes A survey of all the students in your school yields the following probability distribution, where \(X\) is the number of movies that a selected student has seen in the past week: \begin{tabular}{|r|c|c|c|c|c|} \hline Number of Movies & 0 & 1 & 2 & 3 & 4 \\ \hline Probability & \(.5\) & \(.1\) & \(.2\) & \(.1\) & \(.1\) \\ \hline \end{tabular} Compute the expected value \(\mu\) and the standard deviation \(\sigma\) of \(X\). (Round answers to two decimal places.) For what percentage of students is \(X\) within two standard deviations of \(\mu\) ?
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