How to find median

How to Find the Median Value

Median Value

The Median is the «middle» of a sorted list of numbers.

How to Find the Median Value

To find the Median, place the numbers in value order and find the middle.

Example: find the Median of 12, 3 and 5

Put them in order:

The middle is 5, so the median is 5.

Example: 3, 13, 7, 5, 21, 23, 39, 23, 40, 23, 14, 12, 56, 23, 29

When we put those numbers in order we have:

3, 5, 7, 12, 13, 14, 21, 23, 23, 23, 23, 29, 39, 40, 56

There are fifteen numbers. Our middle is the eighth number:

The median value of this set of numbers is 23.

(It doesn’t matter that some numbers are the same in the list.)

Two Numbers in the Middle

BUT, with an even amount of numbers things are slightly different.

In that case we find the middle pair of numbers, and then find the value that is half way between them. This is easily done by adding them together and dividing by two.

Example: 3, 13, 7, 5, 21, 23, 23, 40, 23, 14, 12, 56, 23, 29

When we put those numbers in order we have:

3, 5, 7, 12, 13, 14, 21, 23, 23, 23, 23, 29, 40, 56

There are now fourteen numbers and so we don’t have just one middle number, we have a pair of middle numbers:

In this example the middle numbers are 21 and 23.

To find the value halfway between them, add them together and divide by 2:

21 + 23 = 44
then 44 Г· 2 = 22

So the Median in this example is 22.

Your Turn

Where is the Middle?

A quick way to find the middle:

Count how many numbers, add 1, then divide by 2

Example: There are 45 numbers

45 plus 1 is 46, then divide by 2 and we get 23

So the median is the 23rd number in the sorted list.

Example: There are 66 numbers

66 plus 1 is 67, then divide by 2 and we get 33.5

33 and a half? That means that the 33rd and 34th numbers in the sorted list are the two middle numbers.

So to find the median: add the 33rd and 34th numbers together and divide by 2.

Find median from Data Stream

In this post we are gonna discuss how to find median in a stream of running integers

Median is the middle value in an ordered integer list. If the size of the list is even, there is no middle value. So the median is the mean of the two middle value.

[2,3,4], the median is 3

[2,3], the median is (2 + 3) / 2 = 2.5

Design a data structure that supports the following two operations:

void addNum(int num) — Add a integer number from the data stream to the data structure.

double findMedian() — Return the median of all elements so far.

Step by step solution to the problem :

First solution which comes to our mind for this problem is keeping an sorted array and whenever a new element comes put that in it’s correct position in the sorted array. In this case the median of the sorted array can be our result.

The time complexity of the find() in above approach will be O(1) but while adding each time we have to increase the array size by one, copy to new array, then find median so it’s quite expensive O(n). Let’s think for sometime can we do better…….

Heaps can rescue us in this situation. Idea is somehow if we can divide input numbers at every point into two half such that upper contain elements larger than lower and both the half are in sorted order, with a condition that absolute value of (no of elements in upper-no of elements in lower ) will never be more than 1. Now there can be 3 cases:

a) no of elements in upper >no of elements in lower then clearly the last element in sorted upper is the median.

b) no of elements in upper Initialization: We can implement upper by using minHeap and lower using MaxHeap.

Time complexity of this is O(logn).

a) if both the heap is empty we are adding first element to minHeap(we can add to maxHeap also).

b) If num is num has no place in minHeap as of now. So it can be placed in maxHeap provided maxQueue.size()≤minQueue.size().

Otherwise we pop the top element maxTop from maxHeap and compare it with num, then place minimum of ( maxTop,num) to maxHeap and maximum of ( maxTop,num) to minHeap.

Otherwise we pop the top element minTop from minHeap and offer to maxHeap and offer num to minHeap.

Time complexity of this is O(1).

a) If both the heap size are equal then median is

b) Otherwise median is at the peak of the heap whose size is more.

How to Find the Mean, Median, and Mode

Exploring some measures of central tendency

Kendra Cherry, MS, is an author and educational consultant focused on helping students learn about psychology.

Emily is a board-certified science editor who has worked with top digital publishing brands like Voices for Biodiversity, Study.com, GoodTherapy, Vox, and Verywell.

Knowing how to find the mean, median, and mode can help you interpret data collected through psychological research. These values provide more insight into what may be considered «normal» or «abnormal» for a specific group of people in terms of cognitive processes or behaviors, for instance.

Because they are all measures of central tendency, psychology students often find it easy to confuse the three. Yet, there are differences in what each one is and how it is found. Here are some useful tips to help you distinguish between these measures, as well as how to calculate mean, median, and mode.

Definition of Mean, Median, and Mode

To understand the differences between the mean, median, and mode, let’s start by defining these three terms.

How to Find the Mean

Take these two steps to calculate the mean:

As an example, imagine that your psychology experiment returned the following number set: 3, 11, 4, 6, 8, 9, 6. To calculate the mean, you first add all the numbers together (3 + 11 + 4 + 6 + 8 + 9 + 6 = 47). Then you divide the total sum by the number of scores used (47 / 7 = 6.7). In this example, the mean or average of the number set is 6.7.

Recap of How to Find the Mean

The mean is calculated by adding all the scores together, then dividing by the number of scores you added.

How to Find the Median

The median is the middle score in the set. To find the median, you take these steps:

As an example, consider this set of numbers: 5, 9, 11, 9, 7. First, you arrange them in numerical order (5, 7, 9, 9, 11). Next, you count how many scores you have (5). Divide the total number of scores by 2 (5 / 2) and you get 2.5.

Since you have an odd number of scores, you round 2.5 up to the number 3. The number in the third position of the data set is the median which, in this case, is 9 (5, 7, 9, 9, 11).

To calculate the median for an even number of scores, imagine that your research revealed this set of data: 2, 5, 1, 4, 2, 7. Your first step is to put them in numerical order (1, 2, 2, 4, 5, 7). Next, add up the total number of scores (6), divide by 2, and you get 3 (6 / 2 = 3).

Go to the score in the third position (1, 2, 2, 4, 5, 7) and add it to the number in the next position (1, 2, 2, 4, 5, 7) to get the sum of the two (2 + 4 = 6). Take 6 and divide it by 2 (the total number of scores you added together), and you get 3. So, the median for this example is 3.

Recap of How to Find the Median

The median is calculated by arranging the scores in numerical order, dividing the total number of scores by two, then rounding that number up if using an odd number of scores to get the position of the median or, if using an even number of scores, by averaging the number in that position and the next position.

How to Find the Mode

Of all the measures, finding the mode requires the least amount of mathematical calculation. Instead, since the mode is simply the most frequently occurring score in a distribution, all you do is look at all your scores and select the most common one.

As an example, consider the following number distribution: 2, 3, 6, 3, 7, 5, 1, 2, 3, 9. The mode of these numbers would be 3 since this is the most frequently occurring number (2, 3, 6, 3, 7, 5, 1, 2, 3, 9).

If no number in a set occurs more than once, there is no mode for that set of data. It’s also possible for a data set to have two modes. This is known as bi-modal distribution.

Bi-modal distribution occurs when there are two numbers that are tied in frequency. For example, consider the following set of numbers: 13, 17, 20, 20, 21, 23, 23, 26, 29, 30. In this set, both 20 and 23 occur twice (13, 17, 20, 20, 21, 23, 23, 26, 29, 30). Therefore, they are both modes.

Recap of How to Find the Mode

To find the mode, you identify the score that occurs most often within the data set. In cases where you have a large number of scores, creating a frequency distribution can be helpful in determining the mode.

Pros and Cons of Mean, Median, and Mode

Each measure of central tendency has its own strengths and weaknesses. Here are a few to consider.

While the mean in math is theoretically neutral, some contend that the use of the mean in psychology can lead to inappropriate conclusions if care is not taken with its application. This is due, in part, to behavior and cognition being both complex and variable in nature.

When to Use Mean, Median, and Mode

How do you determine whether to use the mean, median, or mode when analyzing psychology research? The one you select can depend on the data scores themselves.

If there are no outliers in your data set, the mean may be the best choice in terms of accuracy since it takes into account each individual score and finds the average. Conversely, if outliers exist, the median or mode may be more accurate since the results won’t be skewed.

Also consider what you are trying to measure. Are you looking for the average (the mean), do you want to identify the middle score (the median), or are you looking for the score that appears most often (the mode)? While they are all measures of central tendency, each one looks at this tendency from a slightly different point of view.

An Example of Mean, Median, and Mode in Psychology

Imagine a research study in which psychologists are interested in learning the typical age at which someone might be diagnosed with schizophrenia. To collect this data, they send a questionnaire to mental health providers, asking that they share their patients’ ages upon formal diagnosis.

The responses received indicate that the practitioners’ patients were the following ages:

Using the calculations above, you would find that the mean, median, and mode for this data set are all around 27 years (27.1 years, 27 years, and 27 years respectively). In this case, any of these measures could be used to help you arrive at the typical age of onset.

But what if you had an additional score of 13? In this case, the calculation of the mean would be 25.6, while the median and mode would both be 27. Since the mean includes an outlier, median and mode would be more accurate as they aren’t skewed by this number.

In case you are curious, the National Alliance on Mental Health reports that the average age of schizophrenia onset for men is late teens to early 20s, while women tend to be diagnosed with this condition in their late 20s to early 30s.

A Word From Verywell

Mean, median, and mode all serve a valuable purpose in analyzing psychological data. They all also have their pros and cons. Knowing how to find mean, median, and mode—as well as their strengths and weaknesses—can help you better interpret data collected via psychology research.

Speelman CP, McGann M. How mean is the mean? Front Psychol. 2013;4:451. doi:10.3389/fpsyg.2013.00451

National Alliance on Mental Illness. Schizophrenia.

Hogg RV, McKean JW, Craig AT. Introduction to Mathematical Statistics. Boston: Pearson; 2013.

Median

Median represents the middle value for any group. It is the point at which half the data is more and half the data is less. Median helps to represent a large number of data points with a single data point. The median is the easiest statistical measure to calculate. For calculation of median, the data has to be arranged in ascending order, and then the middlemost data point represents the median of the data.

Further, the calculation of the median depends on the number of data points. For an odd number of data, the median is the middlemost data, and for an even number of data, the median is the average of the two middle values. Let us learn more about median, calculation of median for even-odd number of data points, and median formula in the following sections.

What is Median?

Median is one of the three measures of central tendency. When describing a set of data, the central position of the data set is identified. This is known as the measure of central tendency. The three most common measures of central tendency are mean, median, and mode.

Median Definition

The value of the middle-most observation obtained after arranging the data in ascending order is called the median of the data. Many an instance, it is difficult to consider the complete data for representation, and here median is useful. Among the statistical summary metrics, the median is an easy metric to calculate. Median is also called the Place Average, as the data placed in the middle of a sequence is taken as the median.

Median Example

Let’s consider an example to figure out what is median for a given set of data.

Median Formula

Using the median formula, the middle value of the arranged set of numbers can be calculated. For finding this measure of central tendency, it is necessary to write the components of the group in increasing order. The median formula varies based on the number of observations and whether they are odd or even. The following set of formulas would help in finding the median of the given data.

Median Formula for Ungrouped Data

The following steps are helpful while applying the median formula for ungrouped data.

Median Formula When n is Odd

The median formula of a given set of numbers, say having ‘n’ odd number of observations, can be expressed as:
Median = [(n + 1)/2] th term

Median Formula When n is Even

The median formula of a given set of numbers say having ‘n’ even number of observations, can be expressed as:
Median = [(n/2) th term + ((n/2) + 1) th term]/2

Example: The age of the members of a weekend poker team has been listed below. Find the median of the above set.

Solution:

Step 1: Arrange the data items in ascending order.

Step 2: Count the number of observations. If the number of observations is odd, then we will use the following formula: Median = [(n + 1)/2] th term

Step 3: Calculate the median using the formula.

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

= (7 + 1)/2 th term = 4 th term = 42

Median Formula for Grouped Data

When the data is continuous and in the form of a frequency distribution, the median is calculated through the following sequence of steps.

Now, use the following formula to find the median value.

Application of Median Formula

Let us use the above steps in the following practical illustration to understand the application of the median formula.

Solution: We will follow the given steps to find the median salary.

How to Find Median?

We use a median formula to find the median value of given data. For a set of ungrouped data, we can follow the below-given steps to find the median value.

Example: The height (in centimeters) of the members of a school football team have been listed below.

Find the median of the above set.

Solution:

Step 1:

Arrange the data items in ascending order.

Original set:

Ordered Set:

Step 2:

Count the number of observations.

Number of observations, n = 8

If number of observations is even, then we will use the following formula:

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

Step 3:

Calculate the median using the formula.

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

Median = [(8/2) th term + ((8/2) + 1) th term]/2

= (4 th term + 5 th term)/2

For a set of grouped data, we can follow the following steps to find the median:

When the data is continuous and in the form of a frequency distribution, the median is calculated through the following sequence of steps.

We have seen examples to find median for ungrouped data in the previous section. Here’s an example to calculate median for grouped data.

Example: Calculate the median for the following data:

Solution:

We need to calculate the cumulative frequencies to find the median.

l = 40, f = 35, c = 25, h = 20 \(\)

Using Median formula:

Median of Two Numbers

In an ordered series, the median is the number that is mid-way between the range extremes. It is not usually identical to the mean. Let’s understand how to find the median. For a set of two values, the median will be the same as the mean, or arithmetic average. For example, the numbers 2 and 10, both have a mean and a median of 6. Note that the median is the value at the mid-point of the dataset, not the mid-point of the values. The mean is the arithmetic average: (10 + 2)/2 = 6. What if we add two more numbers, say 3 and 4? The median will be 3.5, but the mean will be (2 + 3 + 4 + 10)/4 = 4.75

Important Notes on Median:

The above content to find the median has been summarized in the form of the following points.

Thinking Out of the Box:

Now it’s time to apply the learned concepts of the median. Here’s a question for you!

Question: Determine the median of the first five whole numbers. In a company, for each of the 10 employees working in a service upgrade process, the number of service upgrades sold are as follows: 34, 26, 30, 21, 25, 12, 18, 20, 19, 15. Find out the median number of service upgrades sold by the 10 employees?

Related Topics on Median:

Examples on Median

Example 1: The age of the players of a soccer team has been listed as <42,40,50,65,35,58,32>. Apply the median formula and find the median of the given set of numbers.

Solution:

To find the median of the given number of sets <42,40,50,65,35,58,32>we will follow the following steps.

Step 1: Arrange the data items in ascending order. Given set: <42,40,50,65,35,58,32>; Arranged set: <32,35,40,42,50,58,65>.

Step 2: Count the number of observations. Here, the number of observations (n) = 7.

Step 3: Now use the median formula when ‘n’ is odd, given as,[Median = <(n + 1)/2>th term]

Step 4: Median = [(7 + 1)/2] th term = (8/2) th term = 4 th term.

Step 5: Note down the 4 th observation from the set <32,35,40,42,50,58,65>. The 4 th observation is 42.

Answer: The median of the set <32,35,40,42,50,58,65>is 42.

Example 2: Find the median of the data set <3, 15, 2, 34, 11, 25>.

Answer: When sorted in ascending order, it becomes <2, 3, 11, 15, 25, 34>.
The total number of observations here is n = 6.
Using the median formula when n is even
Median = [(n/2) th term + ((n/2) + 1) th term]/2
Median = [(6/2) rd term + ((6/2) th term + 1)]/2 = (3 rd term + 4 th term)/2
The 3rd term is 11.
The 4th term is 15.
On substituting the values
Median = (11 + 15)/2 = 26/2 = 13
Answer: The median is 13.

Example 3: Marks scored by a student in different subjects are <98, 64, 76, 91, 44, 81>. Using the median formula, calculate the median of the above-given data.

To find: Median for the given set.
Given: Set of marks: <98, 64, 76, 91, 44, 81>
Arranging the set in ascending order: <44, 64, 76, 81, 91, 98>
Number of observations, n = 6 (even)
Using median formula,
Median = [(n/2) th term + ((n/2) + 1) th term]/2
Median = [(6/2) rd term + ((6/2) th term + 1)]/2 = (3 rd term + 4 th term)/2
The 3rd term is 76
The 4th term is 81
On substituting the values
Median = (76 + 81)/2 = 157/2 = 78.5
Answer: Median of the given-set is 78.5

Example 4: Annie noted the number of cakes she baked every day over the past week. The numbers were: 1, 2, 2, 3, 4, 3, 3. What is the median value of the number of cakes she baked?

Solution:

To find the median value of the number of caked baked by Annie we first arrange the numbers of cakes baked per day in a sequence and then pick the middlemost value.
The original set of the number of cakes baked per day: 1, 2, 2, 3, 4, 3, 3.
The Ordered Set of cakes baked per day: 1, 2, 2, 3, 3, 3, 4.
Count the number of observations(n) = 7.
Since the number of observations is odd, median = middle value i.e. 4 th value. Thus, median = 3.
Answer: The median value of cakes she baked is 3.

Example 5: Raffle tickets were being sold during a Carnival. Jack, a Carnival worker, was taking count of the sales each hour. The number of tickets sold every hour was as follows: 130, 123, 146, 109, 112, 111. What was the median number of tickets sold?

Solution:

To find the median number of raffle tickets sold during the carnival, we arrange the data in a sequence and then pick the middlemost.
The original set of the number of tickets sold every hour during the carnival: 130, 123, 146, 109, 112, 111.
Ordered Set of the number of tickets sold during the carnival: 109, 111, 112, 123, 130, 146.
Number of observations= 6 i.e, even.
Use the median formula for even numbers, Median = <(n/2) + (n/2 + 1)>/2. Thus, median = (3rd observation + 4th observation)/2 = (112 + 123)/2 = 235/2.

Therefore, the median number of tickets sold was 117.5.

How To Find Median?

Process to find the Median

The middle value of a set of numbers is called the median. It is an integral concept in analysing Statistical data. The calculation of the median value varies for an odd or even number of values. First of all, the list of numbers needs to be arranged in the order in an ascending sequence or a descending sequence. If there is an odd list of numbers, the midpoint of the list is the median. However, in the case of an even count of numbers, the two numbers in the middle of the list are considered. The process of calculating the median is a little complex. The examples will help the student to understand better.

What is Median?

The Median represents one of the three important measures of central tendency of grouped or ungrouped data. These three most commonly used central tendency is known as Mean (Arithmetic mean), Median and Mode. In order to describe a set of data, the central position of the data set is identified which is just known as the measurement of central tendency.

In an ascending or descending ordered series, Median is the positionally centre most value of a provided set of data (either grouped or ungrouped) or The median can be also be defined as the value present at the midpoint of a given set of data, not the midpoint of the values itself. It is the point from where half of the data entry is more and half of the data entry is less. For the calculation of the median, all the data is first written in ascending or descending order and then the centre most data point is identified and that will be our median.

How to find the Median of Ungrouped Data?

How to Find the Median with ODD Number of observations?

The formula of the median for a given set of numbers, having the odd number of observations can be expressed as such

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

Let us understand with an Example

The age of seven participants of a golf tournament has been listed below. Find the median of this given set.

Ordered set of data :

As the count of all the observations is seven, i.e. n = 7

By the formula of Median = [(n+1)/2]th term

Median = [(7+1)/2] = 4 th term = 44, hence the median is 44.

How to find the Median with an Even Number of Observations?

The formula of the median for a given set of numbers, having the even numbers of observations can be expressed as such

Median = [n/2] th term

Let us understand with an example

Suppose the set has an even count of numbers for example, 8,1, 3, 5, 22,17,12,13. It is a set of 8 numbers. The list when sorted in ascending order 1, 3, 5, 8, 12, 13, 17, 22.

8 and 12 are the two middle numbers here.

Therefore, adding 8 and 12 and dividing the result by 2 = (8 + 12) / 2

Here, 10 is the median of the given list of numbers. 84 199.

How to find the Median in Maths for Grouped Frequencies?

When we have grouped data, calculating the median becomes a little more complicated. Students must be careful during this calculation. Here we consider the following grouped data table for a set of balls,