Statistics Definitions > Median

**Contents:**

- Definition and Formula.
- Calculation for an
**odd**set of numbers. - Calculation for an
**even**set of numbers. - Average vs. Med.
- Calculation for a Grouped Frequency Distribution.

## 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.

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!*

**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.