Chapter 2: Statistical Tools for Quality Descriptive Statistics (Mean, Median, Standard Deviation)

Abstract:

The mean, median, mode, and standard deviation are statistical tools that can be used to describe a set of data and are useful for statistical analysis: 

Mean: A central tendency value that represents the average of the data 

Median: A central tendency value that estimates the middle of the data 

Mode: A central tendency value that represents the most frequently occurring value in the data 

Standard deviation: A measure of variability that represents the average distance between the data and the mean 

Descriptive statistics are used to summarize, organize, and present data in a way that makes it easier to interpret.

 

They often include measures of central tendency, dispersion, and graphical representations. 

Other statistical tools include: coefficient of variation, interquartile range, pooled variance, skewness and kurtosis, and sum of squares. 

Keywords:

Statistical Tools for Quality, 

Descriptive Statistics, Mean, Median, Standard Deviation

Learning Outcomes 

After undergoing this article you will be able to understand the following 

Statistical Tools for Quality

Descriptive Statistics

Mean

Median

Standard Deviation

Chapter 2: Statistical Tools for Quality

Descriptive Statistics (Mean, Median, Standard Deviation)

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2.1 Introduction to Statistical Tools for Quality

Statistical tools are integral in quality management and control as they help organizations monitor, measure, and improve processes and products. Descriptive statistics, in particular, play a crucial role in summarizing data, identifying patterns, and facilitating informed decision-making. This chapter explores three fundamental descriptive statistical measures—mean, median, and standard deviation—and their applications in quality management.

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2.2 Descriptive Statistics

Descriptive statistics provide a summary of data, offering insights into its central tendency, variability, and distribution. These tools enable quality managers to understand data behavior, detect variations, and identify areas requiring improvement. The three primary measures discussed in this chapter include:

1. Mean (Average)
2. Median (Middle Value)
3. Standard Deviation (Measure of Dispersion)

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2.3 The Mean

Definition:
The mean, or average, is the sum of all data points divided by the total number of data points. It represents the central value of a dataset and is highly sensitive to outliers.

Formula:
For a dataset with observations :

Applications in Quality:

• Process Monitoring: The mean is used in control charts to monitor process stability.
• Benchmarking: It helps determine the average performance level of a product or process.
• Customer Satisfaction: Organizations use mean scores from surveys to assess overall customer satisfaction.

Example:
Consider the diameters (in mm) of 10 manufactured bearings: 10.1, 10.2, 10.0, 10.1, 10.3, 10.2, 10.1, 10.2, 10.0, 10.1.

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2.4 The Median

Definition:
The median is the middle value in a sorted dataset. It divides the data into two equal halves and is less affected by outliers compared to the mean.

Steps to Calculate the Median:

1. Arrange the data in ascending order.
2. If the number of observations () is odd, the median is the middle value.
3. If is even, the median is the average of the two middle values.

Applications in Quality:

• Robust Measure: The median is useful when datasets contain outliers or are skewed.
• Quality Surveys: Median scores from customer feedback can provide a better understanding of central trends when extreme responses are present.

Example:
Consider the same bearing diameters: 10.0, 10.0, 10.1, 10.1, 10.1, 10.1, 10.2, 10.2, 10.2, 10.3.
Since (even), the median is the average of the 5th and 6th values:

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2.5 The Standard Deviation

Definition:
Standard deviation measures the dispersion or spread of data around the mean. It indicates how much individual data points deviate from the average, providing insights into process variability.

Formula:
For a dataset with observations , the standard deviation () is:

Where is the mean.

Applications in Quality:

• Process Capability Analysis: Standard deviation helps assess whether a process meets customer specifications.
• Control Charts: It is used to determine control limits in statistical process control (SPC).
• Risk Assessment: A smaller standard deviation indicates consistent quality, while a larger value highlights potential issues.

Example:
For the bearing diameters, calculate the mean ().

The small standard deviation suggests minimal variability in the manufacturing process.

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2.6 Comparison of Mean, Median, and Standard Deviation

Statistic	Key Feature	Advantages	Disadvantages
Mean	Central value of data	Easy to calculate, widely used	Sensitive to outliers
Median	Middle value of sorted data	Robust to outliers	Less representative for symmetric data
Standard Deviation	Measure of data dispersion	Quantifies variability	Complex to compute manually

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2.7 Role of Descriptive Statistics in Quality Management

• Improved Decision-Making: Descriptive statistics provide insights that guide process improvements and decision-making.
• Data Visualization: Measures like mean and standard deviation are essential for creating control charts, histograms, and Pareto diagrams.
• Benchmarking and Target Setting: They help set realistic quality targets and monitor performance against benchmarks.

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2.8 Conclusion

Descriptive statistics are indispensable tools for quality management. The mean, median, and standard deviation offer insights into data trends, central tendencies, and variability. Their proper application enables organizations to maintain high-quality standards, enhance customer satisfaction, and drive continuous improvement. By leveraging these tools, quality professionals can ensure consistent performance and foster a culture of excellence.

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