# Median vs. Average: Deciphering Statistical Measures

When we encounter statistical data, understanding the measures that represent the data’s central tendency is crucial. Median vs. average, also known as the mean, are two such measures that often lead to confusion. While both provide valuable insights, they are calculated differently and can convey different information about a data set.

## Median vs. Average: Understanding the Concepts

### Key Takeaways

• Median and average are central tendency measures, but they are calculated and interpreted differently.
• The median can be more informative for skewed distributions, while the average is sensitive to outliers.
• Knowing when to use median or average is essential for accurate data analysis.

### Median vs. Average: The Definition

#### What Is the Average

The term average commonly refers to the arithmetic mean, which we calculate by adding up all the numbers in a set and then dividing by the count of those numbers. For example, if we have a set of five numbers: 3, 5, 7, 9, and 11, our average would be:

`(3 + 5 + 7 + 9 + 11) / 5 = 35 / 5 = 7`

This calculation tells us the central value of the data set if all the numbers were distributed evenly.

#### What Is the Median

The median, on the other hand, is the middle number in a set of numbers that has been arranged in ascending order. If we have an odd number of observations, the median is simply the middle number. With an even number of observations, it is the average of the two middle numbers. Using the previous set of numbers: 3, 5, 7, 9, and 11, our median is: `7`

However, if we add another number, say 13, to create an even set: 3, 5, 7, 9, 11, and 13, we find the median by averaging the two middle numbers, 7 and 9:

`(7 + 9) / 2 = 16 / 2 = 8`

The median gives us a value that divides our set into two equal halves, where half the numbers are lower and the other half are higher. It is especially useful when we want to find a central tendency that is not skewed by outliers.

### Median vs. Average: How to Calculate

#### Calculating the Average

To calculate the average, or mean, we sum up all the numbers in a data set and then divide that sum by the count of numbers. Here’s a simple step-by-step guide:

1. Add up all the values
2. Count the numbers in the data set
3. Divide the sum by the count to get the average

For example, if we have a data set of `1, 3, 5, 7`, we add these numbers to get `16`. We then divide `16` by the number of data points, which is `4`, to get an average of `4`.

#### Determining the Median

The median is the middle value of a data set when the numbers are arranged in ascending order. If there is an even number of observations, the median is the average of the two middle numbers. To find the median:

1. Arrange the data points in ascending order.
2. If the count of numbers is odd, the median is the middle number.
3. If the count is even, the median is the average of the two middle numbers.

For instance, in an ordered data set `2, 4, 6, 8, 10`, the median is `6` since it’s the middle number. In the data set `2, 4, 6, 8`, we take the average of `4` and `6`, so the median is `5`.

### Comparative Analysis

#### When to Use Average

We use the average, or mean, when data points are uniformly distributed and there are no outliers to skew the sum. The average is calculated by adding all the numbers in a data set and dividing by the count of numbers. For example, if we want to know the average test score in a class where most students perform similarly, averaging would be the best method.

• Uniform Data: With scores of 80, 82, 85, 88, 90, the average is (80 + 82 + 85 + 88 + 90)/5 = 85
• Stable Data Without Outliers: In a factory, if we’re calculating the average daily output and the numbers are consistent, the average gives us a solid approximation.

#### When to Use Median

We use the median to find the middle value in a set of numbers, which is particularly useful when dealing with skewed distributions or outliers. The median is the number that falls in the middle when the numbers are ordered from smallest to largest. It’s an excellent choice when we want to understand the typical value in a set where outliers may distort the average.

Skewed Distributions: If we’re examining income in a region where there’s a significant income disparity, the median gives us a clearer picture of the typical income.

Incomes \$30,000 \$35,000 \$40,000 \$45,000 \$1,000,000
Median     \$40,000

Data with Outliers: Consider a neighborhood where most homes cost around \$250,000, but there’s one worth \$4 million. The median price reflects the standard better than the average.

Home Prices: \$230,000 \$240,000 \$250,000 \$260,000 \$270,000 \$4,000,000
Median:     \$250,000

## Median vs. Average: Examples

### Examples of Median

• The median income for the neighborhood is higher than the national average.
• The median age of the population in the city has been steadily increasing.
• In statistics, the median is a measure of central tendency.
• The real estate agent explained that the median home price in the area had risen significantly.
• The study found that the median time for recovery from the illness was two weeks.
• The median household size in the suburban area is three people.
• The report highlighted the importance of the median voter in shaping political outcomes.

### Examples of Average

• The average temperature in July is usually around 80 degrees Fahrenheit.
• The average student spends about two hours per day on homework.
• Last year, the company’s profits were above the average for similar businesses in the industry.
• The average lifespan of a domestic cat is 12 to 15 years.
• An average of 200 people attend the annual charity event.
• The average commute time to work in the city is 30 minutes.
• The professor explained the concept of the average as a measure of central tendency in statistics.

## Related Confused Words With Median or Average

### Median vs. Altitude

The term “median” is commonly used in statistics and mathematics to refer to the middle value in a set of numbers when they are arranged in ascending order. It is a measure of central tendency and is particularly useful for identifying the midpoint of a distribution. For example, in the set of numbers 10, 15, 20, 25, and 30, the median is 20.

On the other hand, “altitude” typically refers to the height of an object or location above a specific reference point, such as sea level. In geography and aviation, altitude is an important measure for determining the vertical distance between a point on the Earth’s surface and a defined reference point, often used in the context of aircraft flight, mountain heights, or topographical features.

### Average vs. Mean

Mean and average are often used interchangeably, but they have specific meanings in the context of statistics and mathematics.

The mean of a set of numbers is calculated by adding up all the values in the set and then dividing by the total number of values. It is often referred to as the arithmetic average. For example, to find the mean of the numbers 3, 5, 7, and 11, you would add them together (3 + 5 + 7 + 11 = 26) and then divide by the total number of values (4), resulting in a mean of 6.5.

The term “average” is a broader concept that can refer to different types of averages, including the mean, median, and mode. However, in common usage, “average” often refers to the mean. In statistics, the mean is one of the measures of central tendency, along with the median and mode, and it represents the typical value of a set of numbers.

## Median vs. Average: Practice and Exercises

Instruction:

You will be given math examples and asked to calculate both the Median and Average for each example. After that, you will need to provide the answers and explain the differences between Median and Average.

Question:

1. Calculate the Median and Average for the following set of numbers: 5, 7, 3, 9, 8, 2, 6.
2. Calculate the Median and Average for the following set of numbers: 12, 15, 18, 22, 14, 16, 20, 19.
3. Calculate the Median and Average for the following set of numbers: 10, 10, 10, 10, 10, 20, 20, 5, 5.

1. For the set of numbers 5, 7, 3, 9, 8, 2, 6:
• Median: First, arrange the numbers in ascending order: 2, 3, 5, 6, 7, 8, 9. Since there are 7 numbers, the median is the 4th number, which is 6.
• Average: Add all the numbers together and divide by the total count: (5+7+3+9+8+2+6)/7 = 6.14.

The Median is 6 and the Average is 6.14. The Median represents the middle value, while the Average gives us the typical value of the set.

2. For the set of numbers 12, 15, 18, 22, 14, 16, 20, 19:
• Median: Arrange the numbers in ascending order: 12, 14, 15, 16, 18, 19, 20, 22. There are 8 numbers, so the median is the average of the 4th and 5th numbers, which is (16+18)/2 = 17.
• Average: Add all the numbers together and divide by the total count: (12+15+18+22+14+16+20+19)/8 = 17.

The Median is 17 and the Average is 17. Both the Median and Average are the same in this case.

3. For the set of numbers 10, 10, 10, 10, 10, 20, 20, 5, 5:
• Median: Arrange the numbers in ascending order: 5, 5, 10, 10, 10, 10, 10, 20, 20. There are 9 numbers, so the median is the 5th number, which is 10.
• Average: Add all the numbers together and divide by the total count: (10+10+10+10+10+20+20+5+5)/9 = 11.11.

The Median is 10 and the Average is 11.11. The Median is not affected by extreme values, while the Average is influenced by all values in the set.

What is the difference between mean and median in terms of data interpretation?

The mean, or average, is calculated by adding all numbers in a data set and dividing by the total count of numbers. It represents the central tendency but can be skewed by extremely high or low values. The median, however, is the middle value in an ordered list and gives a better sense of the data’s center when there’s skewness.

Can you provide examples demonstrating when to use median over mean?

We use the median to represent the typical value when dealing with incomes, house prices, or any data prone to extreme values. For example, if a neighborhood has one billionaire and nine middle-class residents, the median income more accurately reflects typical earnings than the mean.

How can I tell if the median or the mean gives a better representation of typical income levels?

If income distribution is symmetrical, the mean might suffice. However, if there’s a significant gap between the wealthy and the rest, the median is usually a truer reflection of the typical income, as it’s not influenced by the extremities.

In what situations would the median be a more reliable measure than the average?

The median is more reliable in distributions with outliers or skewed data. If you’re assessing customer satisfaction and most ratings are high with a few low outliers, the median provides a more accurate account of the typical customer experience.

How does the median value relate to the overall distribution of a data set?

The median splits a data set into two equal parts. In a perfectly symmetrical distribution, the median and mean are the same. However, if the distribution is skewed, the median remains centrally located to the data, offering a more consistent measure of center.

Could you explain why the median might differ from the mean in skewed data?

Skewed data has more values on one side of the scale, pulling the mean towards the tail. For example, a high-income outlier can drastically increase the mean, whereas the median remains unchanged, better indicating the central tendency of the majority of the data.

Last Updated on December 25, 2023