# Median sort

What is Median? Median of a sequence or an array of numbers is the middle element of the sequence when the total number of elements is odd or the average of middle elements when the total number of elements is even, provided the sequence is sorted. When total elements are odd. When total elements are even. How to calculate Median in java Following are the different ways in which you can calculate median in a java program.

Method 1 : Finding the middle element Program is given below. Above program initializes arrays with an odd number of elements and even number of elements and passes these arrays to a method which calculates median. This method firsts sorts the array using sort method of java. Arrays class. After this, it checks if the total number of elements of array is even or odd by taking modulus of total elements with 2 and comparing the remainder with 0.

If the remainder is 0, the number of elements are even else they are odd. If the total elements are oddit retrieves the middle element by dividing the total number of elements by 2. If the total elements are evenit calculates the sum of middle two elements 12 and 15 and divides this sum by 2 to calculate the average of middle elements.

Method 2 : Using Apache Math Library Apache Math library provides many utility classes and methods to calculate mathematical entities and solution to programming problems such as mean, median, matrix operations, complex numbers, probability distribution and many more. This library has a class Median in package org.

This class has a median method which takes an array and returns the median value from these array elements. Thus if you have an integer array, then you need to convert it to an array of double elements.

Usage of this library to calculate median is shown below. Note that now no sorting of array is required and you do not need to check the size of the array and provide different handling as per the total elements being odd or even, the library handles this stuff itself. Apache commons library can be found at this link.

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### MEDIAN function

Accept Terms.In statistics and probability theorythe median is the value separating the higher half from the lower half of a data samplea population or a probability distribution. For a data setit may be thought of as the "middle" value. For example, the basic advantage of the median in describing data compared to the mean often simply described as the "average" is that it is not skewed so much by a small proportion of extremely large or small values, and so it may give a better idea of a "typical" value.

For example, in understanding statistics like household income or assets, which vary greatly, the mean may be skewed by a small number of extremely high or low values. Median incomefor example, may be a better way to suggest what a "typical" income is.

The median of a finite list of numbers is the "middle" number, when those numbers are listed in order from smallest to greatest. If there is an odd number of numbers, the middle one is picked. For example, consider the list of numbers. If there is an even number of observations, then there is no single middle value; the median is then usually defined to be the mean of the two middle values.

In more technical terms, this interprets the median as the fully trimmed mid-range. With this convention, the median can be described in a caseless formula, as follows:. Formally, a median of a population is any value such that at most half of the population is less than the proposed median and at most half is greater than the proposed median. As seen above, medians may not be unique. If each set contains less than half the population, then some of the population is exactly equal to the unique median.

The median is well-defined for any ordered one-dimensional data, and is independent of any distance metric. The median can thus be applied to ranked but not numerical classes e. A geometric medianon the other hand, is defined in any number of dimensions. A related concept, in which the outcome is forced to correspond to a member of the sample, is the medoid.

The median is a special case of other ways of summarising the typical values associated with a statistical distribution : it is the 2nd quartile5th decileand 50th percentile. The median can be used as a measure of location when one attaches reduced importance to extreme values, typically because a distribution is skewedextreme values are not known, or outliers are untrustworthy, i.

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The median is 2 in this case, as is the modeand it might be seen as a better indication of the center than the arithmetic mean of 4, which is larger than all-but-one of the values! However, the widely cited empirical relationship that the mean is shifted "further into the tail" of a distribution than the median is not generally true. As a median is based on the middle data in a set, it is not necessary to know the value of extreme results in order to calculate it.

For example, in a psychology test investigating the time needed to solve a problem, if a small number of people failed to solve the problem at all in the given time a median can still be calculated.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. The dark mode beta is finally here. Change your preferences any time. Stack Overflow for Teams is a private, secure spot for you and your coworkers to find and share information.

The Median of medians approach is very popular in quicksort type partitioning algorithms to yield a fairly good pivot, such that it partitions the array uniformly. Its logic is given in Wikipedia as:. Each of these elements is a median of 5, making it less than 2 other elements and greater than 2 other elements outside the block.

The median of these numbers is 3. So that is the idea. That is, for each set of 5 numbers, you get their median. This is obvious. Now if you get the median of those numbers call it mit is bigger than half of them and smaller than the other half by definition of median! This means that for each of those smaller 5 element groups where m was bigger than its medium, m is bigger also than two other numbers.

Learn more. Explanation of the Median of Medians algorithm Ask Question. Asked 7 years, 6 months ago. Active 2 years, 10 months ago. Viewed 19k times. Can somebody explain it a bit lucidly for me.

I am finding it difficult to understand the logic. SexyBeast SexyBeast 7, 21 21 gold badges 84 84 silver badges bronze badges. Active Oldest Votes. Think of the following set of numbers: 5 2 6 3 1 The median of these numbers is 3. Similar logic for the number of elements m is bigger than. Shahbaz Shahbaz Just another question, how does this method guarantee that this number will be the median? The median is a number which partitions the array into the upper and lower half. So what does this figure signify? Well, the median is in the middle, but m in the text above is not the median of all numbers.Keep in touch and stay productive with Teams and Officeeven when you're working remotely.

Returns the median of the given numbers. The median is the number in the middle of a set of numbers. Number1, number2, Number1 is required, subsequent numbers are optional. If there is an even number of numbers in the set, then MEDIAN calculates the average of the two numbers in the middle. See the second formula in the example. Logical values and text representations of numbers that you type directly into the list of arguments are counted.

If an array or reference argument contains text, logical values, or empty cells, those values are ignored; however, cells with the value zero are included. The three most common measures of central tendency are:.

For example, the average of 2, 3, 3, 5, 7, and 10 is 30 divided by 6, which is 5. For example, the median of 2, 3, 3, 5, 7, and 10 is 4. For example, the mode of 2, 3, 3, 5, 7, and 10 is 3.

For a symmetrical distribution of a group of numbers, these three measures of central tendency are all the same. For a skewed distribution of a group of numbers, they can be different. Copy the example data in the following table, and paste it in cell A1 of a new Excel worksheet. For formulas to show results, select them, press F2, and then press Enter. If you need to, you can adjust the column widths to see all the data.

### Algorithms in a Nutshell by George T. Heineman, Stanley Selkow, Gary Pollice

Median of the 6 numbers in the range A2:A7. Because there are six numbers, the median is the midway point between the third and fourth numbers.

Yes No. Any other feedback? How can we improve? Send No thanks. Thank you for your feedback! It sounds like it might be helpful to connect you to one of our Office support agents. Contact Support.Axis or axes along which the medians are computed. The default is to compute the median along a flattened version of the array.

A sequence of axes is supported since version 1. Alternative output array in which to place the result. It must have the same shape and buffer length as the expected output, but the type of the output will be cast if necessary.

If True, then allow use of memory of input array a for calculations. The input array will be modified by the call to median. This will save memory when you do not need to preserve the contents of the input array. Treat the input as undefined, but it will probably be fully or partially sorted.

Default is False. If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the original arr. A new array holding the result. If the input contains integers or floats smaller than float64then the output data-type is np.

Otherwise, the data-type of the output is the same as that of the input. If out is specified, that array is returned instead. Returns the median of the array elements.

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New in version 1. See also meanpercentile. Previous topic numpy. Last updated on Jul 26, Created using Sphinx 1.To find median, input should be sorted from smallest to largest. If input is not sorted, then we have to first sort and them return middle of that list. Question arises is what if number of elements in list are even? In that case, median is average of two middle elements.

Ask of this problem is to find median of two sorted arrays. For example :. Before going into the post, find a pen and paper and try to work out example. And as I tell in our posts, come up with a method to solve this considering, you have all the time and resources to solve this problem. I mean think of most brute force solution. If question was to find median of one sorted array, how would you solved it? Complexity of this operation is O 1.

Focus back on two sorted arrays. To find median of two sorted arrays in no more simple and O 1 operation. How about merging these two sorted array into one, problem is reduced to find median of one array.

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Although to find median in a sorted array is O 1merge step takes O N operations. Hence, overall complexity would be O N.

Reuse the merge part of Merge sort algorithm to merge two sorted arrays. Start from beginning of two arrays and advance the pointer of array whose current element is smaller than current element of other.

This smaller element is put on to output array which is sorted merge array. Merge will use an additional space to store N elements Note that N is here sum of size of both sorted arrays.

Best part of this method is that it does not consider if size of two arrays is same or different. It works for all size of arrays.

This can be optimized, by counting number of elements, N, in two arrays in advance. With this optimizations, time complexity remains O Nhowever, space complexity reduces to O 1. Optimized version to find median of two sorted arrays. One of the property which leads us to think about binary search is that two arrays are sorted. As explained above, median divides input into two equal parts, so first condition median index m satisfy is a[start. After split at specific indices i and j, how can we derive condition for median, which is left part of array should be equal to right part of array?

As we already know these values for split at i and j, equation becomes. But is this the only condition to satisfy for median? As we know, median is middle of sorted list, we have to guarantee that all elements on left array should be less than elements in right array.

It is must that max of left part is less than min of right part. What is max of left part? It can be either A[i-1] or B[j-1].

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What can be min of right part, it can be either A[i] or B[j].The MEDIAN function sorts through the provided arguments to find the value that falls arithmetically in the middle of the group. If there is an odd number of arguments, the function identifies the middle value in the range as the median value.

If there is an even number of arguments, the function takes the arithmetic mean or the average of the middle two values as the median value. The values supplied as arguments do not need to be in any particular order for the function to work. You can see that in play in the fourth row in the example image below.

These steps detail how to enter the MEDIAN function and arguments using the dialog box for the first example displayed in the above image.

Why is the median value 20? For the first example in the image, since there's an odd number of arguments fivethe median value is calculated by finding the middle number. By default, Excel displays a zero 0 in cells with a zero value, as shown in the example above. How to toggle this option on and off in ExcelExcelExceland Excel :.

Tweet Share Email. Arguments The values supplied as arguments do not need to be in any particular order for the function to work. The maximum number of entries allowed iseach of which needs to be separated by a comma. This argument can contain:. Options for entering the function and its arguments:.

Click on cell G2which is where the results will display. Click the Insert Function button to open the Insert Function dialog box. Choose Statistical in the Category list. The answer 20 should appear in cell G2. The median changes between the first and second examples because a zero was added to cell A3 whereas cell A2 is blank.

The addition of a zero to cell A3 changes the number of arguments passed to the function in cell G3 from five to six - an even number. As a result, the median is calculated by adding the two middle values 12 and 20 together and then dividing by two to find their average This option cannot be turned off in Excel Online. Go to the File tab and click Options.