What is the median?

The median is the number that counts the parts into two equal sets of those that are (strictly) smaller and those that are (strictly) larger than the median. A kind of middle member of a data set. The data does not always have to be different. It is important to mention several characteristics:

  • if the number of data is odd, the median has a value in that set;
  • if the data number is even, the median can take the whole range of values;
  • the arithmetic mean can be equal to, greater than, or less than the median

There is a problem with the median. Apart from the fact that it is not unambiguously defined, there is no clear procedure by which we determine it. There are more efficient ways of calculating it, which does not include sorting numbers. Most data processing tools (such as Excel) and programming languages ​​have a built-in function for calculating the median. It gives more real information about numerical data than the arithmetic mean because it is equal to one of these values ​​or is located between two values.

If the given individual values ​​x1, x2,…, xk, xk + 1,…, xN of the characteristic X are ordered by size, and if N = 2k + 1, i.e. an odd number, the median is equal to the value of the characteristic of the central element, the median is xk + l. If N = 2k (even number), the median is (xk + xk + 1) / 2.

How to calculate and use the median calculator?

The median is a data set’s middle number. Arrange the data points in descending order and get the center number. This is the average. If two numbers are in the center, the median is the average of those two values. It displays the value for which 50% of observations are lower and 50% are higher.

Said, it is the value in the middle of the ordered observations. Enter the observed numbers to compute the median from a set of data. Values must be numerical and separated by commas, spaces, or newlines. Unless you intend to study the variance or standard deviation, you do not need to indicate whether the data is from a population or a sample.

Median formulas

The median is the data value that separates the top and lower halves of a data set.

  • Arrange the data values in ascending order from lowest to highest.
  • The median data value is the value in the center of the set.
  • If two data values are in the midway, the median is the mean of those two values.

When there is an odd number of observations, the formula is:

x_m = x_{\frac{n+1}{2}}

When the number of observations is equal, the formula is as follows:

x_m = \frac{x_{\frac{n}{2}} + x_{n+2}{2}}{2}

Median vs. mode

The “median” value in a list of numbers is the “middle” value. Your numbers must be arranged in numerical order from least to greatest to get it. Therefore you may need to redo your list before you can discover the median. The “mode” value is the most often occurring value. If no number in the list is repeated, the list has no mode.

The median is preferable to the mean since it is unaffected by huge numbers. For nominal or categorical data that cannot sort, the mode is the sole metric available. When our data is skewed, we generally favor the median over the mean (or mode) (i.e., the frequency distribution for our data is skewed). On the other hand, the median better preserves its position and is not as heavily impacted by the skewed numbers.

Median symbol

There is no generally acknowledged standard notation for the median; nevertheless, some writers express the median of a variable x as x͂ or as μ1/2, and occasionally as M. In each of these instances, the usage of these or other median symbols must be expressly stated when they are introduced. The median is a subset of several methods for describing the average values associated with a statistical distribution: the second quartile, fifth decile, and fifty percentile.

How to find the median number using median formulas?

The median formula for a given collection of numbers, say with ‘n’ odd observations, may be written as follows:

Median = [(n + 1)/2]th term

The median formula for a given collection of numbers, say with ‘n’ even number of observations, is as follows:

Median = [(n/2)th term + ((n/2) + 1)th term]/2

How to find the median of a set of numbers?

If the number of data items is even, there will be two numbers in the center. The median is the value that falls halfway between these two values. To get the it, arrange all numbers in ascending order and work your way into the center by crossing out numbers at either end.

If there are a lot of data points, add 1 to the total number of data points and then divide by 2 to determine which data point is the median. This works when it’s an odd number, but when it’s an even number, you’ll receive a decimal response like 7.5. It will be located midway between the seventh and eighth items.

An example for the median

Let the following data represent the results of a measurement:

41 11 29 7 37 1013 17 1009 5 23 31 13 2 19 3

Find the median.

Solution:

The data must first be sorted by size

2 3 5 7 11 13 17 19 23 29 31 37 41 1009 1013

As the number of data is odd, the median is the value of the central data – 19. Repeat the previous account without the value 1013, which we subsequently found to be unreliable. If the data number is even, the median will be the mean of the two central data.

2 3 5 7 11 13 17 19 23 29 31 37 41 1009

In this case, the median is 18.0.

Percentiles

Percentiles divide the distribution of results into 100 parts, i.e. each part contains 1% of the distribution results. They can be any integers from 1 to 100. A particular percentile corresponds to a point on the distribution that gives the corresponding percentage of the result to that point, including that result. For example: if someone has 100IQ, he is on the fiftieth percentile, which means that 50% of the population has the same IQ or less.

In the literature, the term percentile rank is often used as a synonym for percentiles. Since the median is the central result in the distribution, it is equal to the 50th percentile. Percentiles are not the same as percentages. Percentiles represent a certain percentage of results in a distribution that are ranked from higher to lower. For example, if someone’s test result is on the 60th percentile, it means that 60% of the respondents did that test worse, and not that the respondent answered precisely 60% of the questions.

Example for percentiles

First, we sort the values ​​by size:

122, 132, 135, 138, 140, 140, 140, 141, 148, 154, 160

We have a total of 11 values, so the first quartile ¼ * 11 = 2.75, we round to the first larger number, in this case, 3, so the first quartile represents the third value in the series, which is 135 cm. This means that 25% of female students have a height equal to or less than 135cm. Or 75% of students are more than 135cm.

Third quartile: ¾ * 11 = 8.25, so the third quartile represents the ninth value in a row – 148cm. This means that 75% of female students have a height equal to or less than 148cm. Or 25% of female students are over 148cm.

A few more examples of median

  1. The data are:

3, 5, 7, 2, 1, 7, 4, 3, 5, 5.

The average is 4.2, and the median is 4.5. Also if you add the number 50 to the existing sequence, the average will be 8.36, and the median 5. Please choose the number yourself and add it to this sequence and observe how the median changes and how the average changes

2. The data are:

3, 5, 7, 2, 1, 7, 4, 3, 5, 5, 5.

To calculate it, we classify this sequence from the smallest to the largest. Furthermore:

1, 2, 3, 3, 4, 5, 5, 5, 5, 7, 7.

There are 11 data in that sequence (odd number). The sixth position shows it. According to that, the result is number 5.

FAQ

How to calculate the median for even numbers?

If the number is an even number, add the two middles and divide by two. The median will be the outcome.

How to calculate median overall survival?

Round down after dividing the number of topics by two. In the example, 5 2 = 2.5, and rounding down to 2 yields 2. Find the first-ordered survival time that exceeds this value. This is the average length of survival.

Median calculation formula

The median formula is (n + 1) 2nd, where “n” refers to the number of elements in the collection and “th” refers to the (n) number. To get the median, first, arrange the numbers in ascending order from smallest to greatest. Then locate the number in the center.

Other calculators

Our Mean Median Mode Calculator is the simplest method to calculate the mean, median, mode, and other quantities. That calculator not only gives you the result but also walks you through the processes to teach you how to calculate the mean, median, and mode. For more calculators in math, physics, finance, health, and more, visit our CalCon Calculator official page.