When a group of numbers is sorted, the median is the number that falls in the middle. It is a measure of central tendency that distinguishes between the minimum and maximum values.
Methods for determining the middle value of a set of data vary according to whether the number of observations is odd or even. The median is defined as the average of the middle two figures in a set of data.
While the median is most often calculated for numerical datasets, it can also be calculated for ordinal datasets (where the values are ranked categories).
What is the Median?
If you have a group of numbers and want to know where the middle is, you can use a statistic called the median. It has practical applications in many areas, such as bankruptcy law, which requires that filers have incomes below the state median before they are eligible for bankruptcy protection.
The median formula is {(n + 1) ÷ 2}th, where “n” is the number of items in the set and “th” just means the (n)the 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, and 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.
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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
When describing a group of numbers where outliers can skew the average, like salaries, the median is a great middle ground. As of the end of 2012, the median income in the United States was $51,017. These American billionaires could artificially inflate the average if their wealth were included.
Suppose you were interested in joining a modest 11-person law practise that paid its employees an average of almost $73,000 per year. You might assume that getting a high-paying job is rather easy. But here’s a look at the math behind determining the mean for those eleven workers:
| 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?
Finding the median, or the middle wage, is a more accurate way to characterize financial well-being. A more accurate depiction of income might be obtained by finding the median of the same set of incomes. The median is the midway value; for example, if one were to arrange all of the earnings in a list (from lowest to highest), they would yield:
$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.
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The 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:
Excel’s median can be calculated using either a built-in function or the Data Analysis tool. When compared to manually entering formulas, using the Data Analysis Toolpak offers numerous benefits. For starters, it has a tonne of extra features that you won’t find in Excel’s base installation (like histograms). Furthermore, Toolpak’s Click and Go interface means you won’t have to commit any formulas to memory.
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.”
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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.” The 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.”
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Median in Minitab (video)
This video shows you how to find the median in Minitab in under a minute:
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