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Chapter 1: Problem 32

A factory quality control manager decides to investigate the percentage of defective items produced each day. Within a given work week (Monday through Friday) the percentage of defective items produced was \(2 \%, 1.4 \%, 4 \%, 3 \%, 2.2 \%\). (a) Calculate the mean for these data. (b) Calculate the standard deviation for these data, showing each step in detail.

Short Answer

Expert verified
The mean is 2.52% and the standard deviation is approximately 0.899%.

Step by step solution

01

Convert Percentages to Decimals

The defective percentages are given as percentages, so first, convert them to decimals for easier calculations. Thus, the values are: - Monday: 0.02 (2%) - Tuesday: 0.014 (1.4%) - Wednesday: 0.04 (4%) - Thursday: 0.03 (3%) - Friday: 0.022 (2.2%) These conversions will be used for both the mean and standard deviation calculation.
02

Calculate the Mean

To find the mean percentage of defective items, sum up the decimal values from Step 1 and then divide by the number of days:\[\text{Mean} = \frac{0.02 + 0.014 + 0.04 + 0.03 + 0.022}{5}\]Calculate the numerator:\[0.02 + 0.014 + 0.04 + 0.03 + 0.022 = 0.126\]Now divide by 5:\[\text{Mean} = \frac{0.126}{5} = 0.0252\]Therefore, the mean percentage is 2.52%.
03

Calculate the Deviations from the Mean

Subtract the mean from each day's decimal to get the deviations:- Monday: \(0.02 - 0.0252 = -0.0052\)- Tuesday: \(0.014 - 0.0252 = -0.0112\)- Wednesday: \(0.04 - 0.0252 = 0.0148\)- Thursday: \(0.03 - 0.0252 = 0.0048\)- Friday: \(0.022 - 0.0252 = -0.0032\)
04

Calculate the Squared Deviations

Square each deviation calculated in Step 3:- Monday: \((-0.0052)^2 = 0.00002704\)- Tuesday: \((-0.0112)^2 = 0.00012544\)- Wednesday: \((0.0148)^2 = 0.00021904\)- Thursday: \((0.0048)^2 = 0.00002304\)- Friday: \((-0.0032)^2 = 0.00001024\)
05

Find Variance by Calculating the Average of Squared Deviations

To find the variance, sum up the squared deviations and divide by the number of observations (5 days):\[\text{Variance} = \frac{0.00002704 + 0.00012544 + 0.00021904 + 0.00002304 + 0.00001024}{5}\]Calculate the numerator:\[0.00002704 + 0.00012544 + 0.00021904 + 0.00002304 + 0.00001024 = 0.0004048\]Now divide by 5:\[\text{Variance} = \frac{0.0004048}{5} = 0.00008096\]
06

Calculate the Standard Deviation

The standard deviation is the square root of the variance:\[\text{Standard Deviation} = \sqrt{0.00008096} \approx 0.00899\]To express this as a percentage, multiply by 100:\[0.00899 \times 100 \approx 0.899\%\]Therefore, the standard deviation is approximately 0.899%.

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Key Concepts

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

Mean
The mean is one of the central concepts in statistics and refers to the average of a set of numbers. To calculate the mean of a data set, you add together all of the data points and then divide by the number of data points. This gives you a simple measure of the central tendency of the data.
In our defective item example, after converting percentages to decimals, we have values for each day of the workweek. To find the mean percentage of defective items, we sum these values: 0.02, 0.014, 0.04, 0.03, and 0.022, totaling 0.126. We then divide this sum by 5 (the number of days), giving us the mean which is 0.0252 or 2.52% when converted back to a percentage.
This calculation helps us understand the general level of defectiveness across the week.
Standard Deviation
The standard deviation measures how much the data varies from the mean. It is a key concept in descriptive statistics that tells us about the spread or dispersion of a dataset.
To find the standard deviation:
  • Calculate the mean.
  • Find the difference between each data point and the mean.
  • Square these differences to remove negative values.
  • Calculate the mean of these squared differences; this result is called the variance.
  • Take the square root of the variance to get the standard deviation.
For our data, the standard deviation is calculated by first finding each deviation from the mean and then squaring these deviations. After summing and averaging these squares, we found the variance to be approximately 0.00008096. Finally, the square root of the variance gives us a standard deviation of about 0.00899, or 0.899% when expressed in percentage terms.
This value tells us how much variation there is from the mean percentage of defective items.
Variance
Variance gives us an idea of how data points in a data set spread out around the mean. It is crucial for understanding the degree of scatter within data.
In our example, after finding the differences between each daily percentage and the mean, we square these deviations: this yields positive numbers regardless of whether the original deviation was negative or positive. By summing these squared deviations and dividing by the number of observations, we obtain the variance.
In our specific case, the variance is 0.00008096. This tells us, in squared units, how far away the data points are from the mean. While it might seem difficult to interpret directly, knowing the variance is foundational for calculating and understanding the standard deviation.
Data Analysis
Data analysis involves examining, cleaning, transforming, and modeling data to extract useful information and support decision-making. The concepts of mean, variance, and standard deviation are all vital tools in this process.
In evaluating the performance of a production process, descriptive statistics allow us to measure consistency and quality trends. In the factory example, calculating these statistics helps managers assess the quality control and determine whether variations in defect rates are within an acceptable range.
By understanding the central tendency (mean) and dispersion (variance and standard deviation) of defective percentages, managers can make informed decisions about necessary adjustments in the manufacturing process.
Ultimately, data analysis aids in identifying potential problem areas, predicting future performance, and ensuring the output quality meets set standards.

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