# Guide: Mean, Median and Mode

In Lean Six Sigma an important part is conducting data and statistical analysis, the concepts of Mean, Median and Mode are some of the basic fundamentals that need to be understood by someone holding a lean six sigma qualification as they are the building blocks and stepping stones to analysis such as histograms and standard deviation.

## Table of Contents

## Mean (Arithmetic Average)

The term “mean” is commonly referred to as the average as it is the average of a set of numbers. However it is important to distinguish it as one of the averages, Mean, Median and Mode are all types of averages.

The Mean average it’s what you get when you add up all the numbers in a data set and then divide that sum by the number of items in the set. The mean gives us a single value that represents the ‘central tendency’ of a data set, helping us understand what a “typical” value might look like.

### Mathematical Formula

The mathematical formula to calculate the mean Mean(x̄) is:

### How to Calculate the Mean

- Write down all the values in your data set.
- Sum up all the values to get a total.
- Count how many numbers are in the data set.
- Take the sum from step 2 and divide it by the count from step 3.

### Example with Data

Imagine a factory produces 5 batches of widgets daily, and the number of widgets in each batch are as follows: 20, 25, 22, 24, 21.

**List all the numbers**: 20, 25, 22, 24, 21**Add them up**: 20+25+22+24+21=112**Count the numbers**: There are 5 batches.**Divide**: 112 / 5 = 22.4

So, the mean number of widgets produced in each batch is 22.4.

### Uses in Lean Six Sigma

In Lean Six Sigma, the mean can be used to measure the central tendency of a process over time. It’s often used as a baseline for comparisons and to identify outliers or trends in data. as an example, if the mean time to manufacture a widget is 20 minutes, any significant deviation from this could indicate an issue that needs investigating.

## Median

The “median” is the middle number in a set of numbers arranged in ascending order. If the data set has an odd number of observations, the median is the number that sits exactly in the middle. If the set has an even number of numbers, the median is the average of the two middle numbers. Unlike the mean, the median is not affected by outliers or skewed distributions, making it a reliable measure of central tendency in certain situations.

### Mathematical Formula

There isn’t a specific formula to calculate the median like there is for the mean. However, the process identifying the median can be explained as:

### How to Calculate the Median

- Arrange all the numbers in your data set in ascending order.
- If the set has an odd number of observations, the middle number is your median. If the set has an even number, take the two middle numbers.
- If you had to take two middle numbers (even set), add them together and divide by 2 to find the median.

### Example with Data

for example if a logistics company has recorded the delivery times for a week as follows: 40, 55, 45, 43, 38, 60, 50.

**Sort the Data**: 38, 40, 43, 45, 50, 55, 60**Find the Middle**: The data set has 7 observations (odd), so the middle number is 45.**Calculate**: The median is 45 minutes.

### Uses in Lean Six Sigma

In Lean Six Sigma, the median is useful when you’re dealing with skewed data or outliers that could distort the mean. For example, if you are measuring the time it takes to resolve customer service tickets and one ticket took an exceptionally long time to resolve, using the median can give you a more accurate picture of typical resolution times than is you used the mean.

## Mode

The “mode” is the number that appears most frequently in a data set and is commonly remembered as “the mode is the most”. Unlike the mean and median, a data set can have more than one mode if multiple numbers appear the most are of an equal frequency. If no number repeats, the data set is said to have no mode. The mode is especially useful when you want to identify the most common occurrences in a given data set.

### How to Calculate the Mode

- Write down all the numbers in your data set.
- Count how many times each number appears.
- The number(s) that appear most frequently is the mode.

### Example with Data

Let’s say a warehouse is tracking the number of boxes processed by each worker in a day. The counts for one day are: 10, 12, 12, 14, 10, 15, 14, 12.

**List the Numbers**: 10, 12, 12, 14, 10, 15, 14, 12**Count the Frequency**:- 10 appears 2 times
- 12 appears 3 times
- 14 appears 2 times
- 15 appears 1 time

**Identify the Mode**: The number 12 appears most frequently, so it is the mode.

### Uses in Lean Six Sigma

The mode can serve as an indicator of the most common outcome or behavior in a process. For instance, in a manufacturing setting, the mode can help identify the most frequent cause of product defects.

In customer service, it can identify the most common types of queries or complaints, helping to target improvements more effectively.

## Comparing Mean, Median, and Mode

Understanding when to use the mean, median, or mode is important as picking the wrong measure for the type of data our the output you want can give you an incorrect analysis of your data.

Each of these measures of central tendency has its own strengths and weaknesses, and choosing the right one depends on the type of data you’re dealing with and the specific problem you’re trying to solve. To help you understand which measure to use when consider the table below which compares the Mean, Median and Mode.

### Tabular Comparison of the Three Measures

Measure | Definition | Mathematical Formula | Advantages | Limitations |
---|---|---|---|---|

Mean | Average of all numbers in a data set | $\frac{\text{Sumofallvalues}}{\text{Totalnumberofvalues}}$ | Easy to compute, suitable for further statistical analysis | Sensitive to outliers, skewed data |

Median | Middle number in a sorted data set | Varies based on even or odd number of observations | Not sensitive to outliers, represents skewed data better | Not suitable for further statistical analysis |

Mode | Number that appears most frequently in a data set | Counting frequency of each number | Easy to understand, identifies most frequent occurrences | May have multiple modes or none, not useful numerically |

### When to Use Each Measure

**Mean**

**When**: You have a large data set that is normally distributed and not skewed by outliers.**Examples**: Measuring average cycle times, average customer satisfaction scores.

**Median**

**When**: Your data set is small, skewed, or contains outliers.**Examples**: Measuring delivery times, resolution times for customer service tickets.

**Mode**

**When**: You want to identify the most common occurrence or category within a data set.**Examples**: Identifying most frequent defects, most common customer complaints.

## Conclusion

Understanding the concepts of mean, median, and mode is an important skill that is needed for basic data analysis before progressing on to more in depth statistical analysis which can significantly impact the success of your Lean Six Sigma projects and continuous improvement efforts.

Each of these measures serves a unique purpose and is best suited for specific types of data and problem statements. The mean offers a general overview but can be skewed by outliers. The median provides a more “typical” snapshot, especially when data is skewed or contains outliers. The mode, on the other hand, highlights the most frequent occurrences, helping you identify patterns. Choosing the right measure of central tendency can make the difference between an accurate analysis and a misleading one.

## References

- Runnenburg, J.T., 1978. Mean, median, mode.
*Statistica Neerlandica*,*32*(2), pp.73-79. - Kaas, R. and Buhrman, J.M., 1980. Mean, median and mode in binomial distributions.
*Statistica Neerlandica*,*34*(1), pp.13-18.

## Author

#### Daniel Croft

Daniel Croft is a seasoned continuous improvement manager with a Black Belt in Lean Six Sigma. With over 10 years of real-world application experience across diverse sectors, Daniel has a passion for optimizing processes and fostering a culture of efficiency. He's not just a practitioner but also an avid learner, constantly seeking to expand his knowledge. Outside of his professional life, Daniel has a keen Investing, statistics and knowledge-sharing, which led him to create the website learnleansigma.com, a platform dedicated to Lean Six Sigma and process improvement insights.

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