Statistics How To

Median: Definition, How to Find it, Formula

Statistics Definitions > Median

Contents:

  1. Definition and Formula.
  2. Calculation for an odd set of numbers.
  3. Calculation for an even set of numbers.
  4. Average vs. Med.
  5. Calculation for a Grouped Frequency Distribution.
  6. Find the median in Excel

What is the Median?

The median is a number in statistics that tells you where the middle of a data set is. It’s used for many real-life situations, like Bankruptcy law, where you can only claim bankruptcy if you are below the median income in their state.
basic statistics median
The median formula is {(n + 1) ÷ 2}th, where “n” is the number of items in the set and “th” just means the (n)th number.
To find the median, you must first order the numbers from smallest to largest. Then find the middle number. For example, the middle for this set of numbers is 5, because 5 is right in the middle:
1, 2, 3, 5, 6, 7, 9.
You get the same result with the formula. There are 7 numbers in the set, so n=7:
{(7 + 1) ÷ 2}th
= {(8) ÷ 2}th
= {4}th
The 4th number in 1, 2, 3, 5, 6, 7, 9 is 5.

However: A caution with using the median formula: The steps differ slightly depending on whether you have an even or odd amount of numbers in your data set.

Find the median for an odd set of numbers

Sample question: Find the median for the following data set:
102, 56, 34, 99, 89, 101, 10.

Step 1: Place the data in ascending order. In other words, sort your data from the smallest number to the highest number. For this sample data set, the order is:
10, 34, 56, 89, 99, 101, 102.

Step 2: Find the number in the middle (where there are an equal number of data points above and below the number):
10, 34, 56, 89, 99, 101, 102.
The median is 89.

Tip: If you have a large data set, divide the number in the set by two. That tells you how many numbers should be above and how many numbers should be below. For example, 101/2 = 55.5. Ignore the decimal: 55 numbers should be above and 55 below.

Find the median for an even set of numbers

Sample question: Find the median for the following data set:
102, 56, 34, 99, 89, 101, 10, 54.

Step 1: Place the data in ascending order (smallest to highest).
10, 34, 54, 56, 89, 99, 101, 102.

Step 2: Find the TWO numbers in the middle (where there are an equal number of data points above and below the two middle numbers).
10, 34, 54, 56, 89, 99, 101, 102

Step 3: Add the two middle numbers and then divide by two, to get the average:
56 + 89 = 145
145 / 2 = 72.5.
The median is 72.5.
Tip: For large data sets, divide the number of items by 2, then subtract 1 to find the number that should be above and the number that should be below. For example, 100/2 = 50. 50 – 1 = 49. The middle two numbers will have 49 items above and 49 below.
That’s it!

Average vs. Median

The median is very useful for describing things like salaries, where large figures can throw off the mean (the average). The median salary in the U.S. as of 2012 was $51,017. If an average was used, those American billionaires could skew that figure upwards.

Let’s say you wanted to work for a small law firm that paid an average salary of over $73,000 to its 11 employees. You might think there’s a good chance you’ll land a great paying job. But take a closer look at how the average is calculated for those eleven employees:

Employee Salary
Samuel $28,000
Candice $17,400
Thomas $22,000
Ted $300,000
Carly $300,000
Shawanna $20,500
Chan $18,500
Janine $27,000
Barbara $21,000
Anna $29,000
Jim $20,000

Average (Mean) =
($28,000 + $17,400 + $22,000 + $300,000 + $300,000 + $20,500 + $18,500 + $27,000 + $21,000 + $29,000 + $20,000) / 11 = $73,000

The two partners in the firm — Ted and Carly, have increased the average way beyond most of the salaries paid in the firm.

See how the “average” can be misleading?

A better way to describe income is to figure out the median — or the middle wage. If you took that same list of incomes and found the median, you would get a more realistic representation of income. The median is the middle number, so if you placed all of the incomes in a list (from smallest to largest) you would get:

$17,400, $18,500, $20,000, $20,500 $21,000, $22,000, $27,000, $28,000, $29,000, $300,000, $300,000

It’s a more accurate representation of what people are actually being paid.

Calculation for a Grouped Frequency Distribution

An easy way to ballpark the median(MD) for a grouped frequency distribution is to use the midpoint of the interval. If you need something more precise, use the formula:
MD = lower value + (B ÷ D) x C.

Step 1: Use (n + 1) / 2 to find out which interval has the MD. For example, if you have 11 intervals, then the MD is in the sixth interval: (11 + 1) / 2 = 12 / 2 = 6. This interval is called the MD group.

Step 2: Calculate “A”: the cumulative percentage for the interval immediately before the median group.

Step 3: Calculate “B”: subtract your step 2 value from 50%. For example, if the cumulative percentage is 45%, then B is 50% – 45% = 65%.

Step 4: Find “C”: the range (how many numbers are in the interval).

Step 5: Find “D”: the percentage for the median interval.

Step 7: Find the median: Median = lower value + (B ÷ D) x C.

That’s it!

Find the median in Excel

Contents:

  1. Excel 2013.
  2. Excel 2007-2010.

Excel 2013

Watch the video or read the steps below:

There are two ways to find the median in Excel — with a function or with the Data Analysis tool. The Data Analysis Toolpak (How to load the Data Analysis Toolpak) has several advantages over just entering formulas. First, it gives you access to many functions that aren’t available in the standard package of Excel (like histograms). Second, you won’t have to remember formulas, as the Toolpak is just a Click and Go interface.

MEDIAN function

Step 1: Type “=MEDIAN(A1:A12)” into a blank cell where “A1:A12” is the location of your data. For example, if you’ve typed your data into D1 to D12, change to “=MEDIAN(D1:D12)”.
Step 2: Press “Enter.”

Data Analysis Toolpak

Step 1: Click the “Data” tab and then click “Data Analysis.”
Step 2: Click “Descriptive Statistics” and then click “OK.”
Step 3: Click the Input Range box and then type the location for your data. For example, if you typed your data into cells A1 to A10, type “A1:A10” into that box
Step 4: Click the radio button for Rows or Columns, depending on how your data is laid out.
Step 5: Click the “Labels in first row” box if your data has column headers.
Step 6: Click the “Descriptive Statistics” check box.
Step 7: Select a location for your output. For example, click the “New Worksheet” radio button.
Step 8: Click “OK.”

Median in Excel 2007-2010.


Sample question: Find the median for the following sample: 123,563,567,22,498,593,947,4,46,876,223,567.1,222.22.

Step 1: Enter your data into a single column. For this example, type “123” into cell A1, press “Enter” and continue entering the numbers down the column from A1 to A13.

Step 2: Click an empty cell.

Step 3: Click the “Formulas” tab and then click “Insert Function.”

Step 4: Type “Median” into the ‘Search for a function’ text box and then click “Go.” Median should be highlighted in the results list. Click “OK.”

Step 5: Type the cell range into the “Number1” cell. In most cases, Excel will automatically populate this with your list already. If it doesn’t, type “A1:A13” into the Number1 box, where “A1:A13” is the actual location of your data.

Step 6: Click “OK.” The answer will be displayed in the cell you selected in Step 2. For this sample, the median is 498.

To Place Your Data in Order:

  1. Click the letter at the top of the column where you want ordered numbers. For this example, click “A.”
  2. Click the arrows to the right of the drop down lists to change your options. For example, you might choose Order as “Smallest to Largest” or “Largest to Smallest.”
  3. Click “OK.”

Check out our YouTube channel for more stats help and tips!

Next: Median in Minitab

------------------------------------------------------------------------------

If you prefer an online interactive environment to learn R and statistics, this free R Tutorial by Datacamp is a great way to get started. If you're are somewhat comfortable with R and are interested in going deeper into Statistics, try this Statistics with R track.

Comments are now closed for this post. Need help or want to post a correction? Please post a comment on our Facebook page and I'll do my best to help!
Median: Definition, How to Find it, Formula was last modified: October 26th, 2017 by Stephanie Glen

One thought on “Median: Definition, How to Find it, Formula