In these lessons, we will learn:

- how to find the median of a frequency table when the number of observations is odd.
- how to find the median of a frequency table when the number of observations is even.
- how to find the median for both discrete and grouped data.

**Related Pages**

Median

Mean And Mode From The Frequency Table

Central Tendency

More Statistics Lessons

The median is the **middle value** in an ordered set of data.

In a frequency table, the observations are already arranged in an ascending order. We can obtain the median by looking for the value in the middle position.

If there is an odd number of observations, the median is the middle number.

If there is an even number of observations, the median will be the mean of the two central numbers.

The following table shows how to find the median from the frequency table with odd number of observations and with even number of observations. Scroll down the page for examples and step-by-step solutions.

** Case 1.** When the number of observations (n) is odd, then
the median is the value at the position.

**Example:**

The following is a frequency table of the score obtained in a mathematics quiz. Find the median score.

Score | 0 | 1 | 2 | 3 | 4 |
---|---|---|---|---|---|

Frequency | 3 | 4 | 7 | 6 | 3 |

**Solution:**

Number of scores = 3 + 4 + 7 + 6 + 3 = 23 (odd number)

Since the number of scores is odd, the median is at the position.

To find out the 12 ^{th} position, we need to add up the frequencies as shown:

Score | 0 | 1 | 2 | 3 | 4 |
---|---|---|---|---|---|

Frequency | 3 | 4 | 7 | 6 | 3 |

Position | 3 | 3 + 4 = 7 | 7 + 7 =14 |

The 12^{th} position is after the 7^{th} position but before the 14^{th}
position. So, the median is 2.

** Case 2.** When the number of observations (n) is even, then
the median is the average of values at the n/2 and (n/2 + 1) positions.

**Example:**

The table is a frequency table of the scores obtained in a competition. Find the median score.

Scores | 0 | 1 | 2 | 3 | 4 |
---|---|---|---|---|---|

Frequency | 11 | 9 | 5 | 10 | 15 |

**Solution:**

Number of scores = 11 + 9 + 5 + 10 + 15 = 50 (even number)

Since the number of scores is even, the median is at the average of the position and position.

To find out the 25^{th} position and 26^{th} position, we add up the frequencies as
shown:

Scores | 0 | 1 | 2 | 3 | 4 |
---|---|---|---|---|---|

Frequency | 11 | 9 | 5 | 10 | 15 |

Position | 11 | 11 + 9 = 20 | 20 + 5 = 25 | 25 + 10 = 35 | 36 to 50 |

The score at the 25^{th} position is 2 and the score at the 26^{th} position is 3.

The median is the average of the scores at 25^{th} and 26^{th} positions =

**How to find the Mean, Median and Mode from a frequency distribution table?**

**Example:**

The one hundred families in a particular neighborhood are asked their annual household income, to the
nearest $5 thousand dollars. The results are summarized in a frequency table. Find the median household
income.

Try the free Mathway calculator and
problem solver below to practice various math topics. Try the given examples, or type in your own
problem and check your answer with the step-by-step explanations.

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