How to Find the Median of a Data Set
Contents:
- Definition and Median Formula.
- Calculation for an odd set of numbers.
- Calculation for an even set of numbers.
- Average vs. Median.
- Calculation for a Grouped Frequency Distribution.
- Excel Instructions
- Median in Minitab (video)
Watch the video to see how to calculate the median:
Can't see the video? Click here.
What is the Median?
The median 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 your state.
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, 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.
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
Example question: Find the median for the following data set:
102, 56, 34, 99, 89, 101, 10.
Step 1: Sort your data from the smallest number to the highest number. For this example 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 2. 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
Example 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 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:
- Excel 2013.
- 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 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 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.
Example question: Find the median for the following set: 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:
- Click the letter at the top of the column where you want ordered numbers. For this example, click "A."
- 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."
- Click "OK."
Median in Minitab (video)
This video shows you how to find the median in Minitab in under a minute:
Can't see the video? Click here.
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How to Find the Median of a Data Set
Source: https://www.statisticshowto.com/probability-and-statistics/statistics-definitions/median/