How To Find The Median Between Two Numbers

A measure of central tendency is a summary statistic that represents the center point or typical value of a dataset. These measures indicate where near values in a distribution fall and are besides referred to as the key location of a distribution. You can think of it as the tendency of data to cluster effectually a middle value. In statistics, the three nigh mutual measures of central tendency are the mean, median, and style. Each of these measures calculates the location of the central bespeak using a unlike method.

Choosing the best measure of central tendency depends on the type of information you accept. In this post, I explore these measures of central trend, show you how to calculate them, and how to determine which one is best for your data.

Locating the Heart of Your Information

Most articles that you'll read about the mean, median, and manner focus on how you calculate each i. I'm going to take a slightly dissimilar approach to start out. My philosophy throughout my web log is to assistance you intuitively grasp statistics by focusing on concepts. Consequently, I'grand going to start by illustrating the central point of several datasets graphically—so you sympathize the goal. So, we'll move on to choosing the best measure out of central tendency for your data and the calculations.

The three distributions below represent different data atmospheric condition. In each distribution, look for the region where the almost common values fall. Even though the shapes and type of data are dissimilar, yous can discover that primal location. That's the expanse in the distribution where the most common values are located.

As the graphs highlight, y'all can encounter where most values tend to occur. That'due south the concept. Measures of primal tendency represent this idea with a value. Coming up, you lot'll acquire that as the distribution and kind of data changes, and so does the all-time measure of cardinal tendency. Consequently, you demand to know the type of data y'all have, and graph information technology, earlier choosing a measure of key trend!

Related posts: Guide to Information Types and How to Graph Them

The central trend of a distribution represents one characteristic of a distribution. Another attribute is the variability around that central value. While measures of variability is the topic of a unlike commodity (link below), this property describes how far away the data points tend to fall from the center. The graph below shows how distributions with the same central tendency (hateful = 100) can actually be quite dissimilar. The panel on the left displays a distribution that is tightly clustered around the mean, while the distribution on the correct is more than spread out. It is crucial to understand that the primal trend summarizes only one aspect of a distribution and that information technology provides an incomplete picture by itself.

Related mail: Measures of Variability: Range, Interquartile Range, Variance, and Standard Deviation

Mean

The mean is the arithmetic boilerplate, and it is probably the measure of central tendency that you are nigh familiar. Calculating the mean is very uncomplicated. You just add together up all of the values and divide past the number of observations in your dataset.

$\frac {x_{1}+x_{2}+\cdots +x_{n}}{n}$

The adding of the mean incorporates all values in the information. If y'all change any value, the mean changes. Even so, the mean doesn't always locate the center of the information accurately. Observe the histograms beneath where I brandish the mean in the distributions.

In a symmetric distribution, the hateful locates the heart accurately.

However, in a skewed distribution, the mean can miss the mark. In the histogram above, it is starting to autumn exterior the key expanse. This problem occurs because outliers have a substantial impact on the mean. Extreme values in an extended tail pull the mean abroad from the center. Every bit the distribution becomes more than skewed, the mean is drawn further abroad from the center. Consequently, it'due south all-time to use the hateful as a mensurate of the central tendency when yous take a symmetric distribution.

In statistics, nosotros generally employ the arithmetics mean, which is the type I discuss in this post. Still, at that place are other types of means, such every bit the geometric mean. Read my mail near the geometric mean to learn when information technology is a better measure.

When to use the mean: Symmetric distribution, Continuous data

Related posts: Using Histograms to Understand Your Data and What is the Hateful?

Median

The median is the middle value. It is the value that splits the dataset in one-half. To find the median, order your information from smallest to largest, and and so find the data point that has an equal amount of values above it and below information technology. The method for locating the median varies slightly depending on whether your dataset has an even or odd number of values. I'll prove y'all how to notice the median for both cases. In the examples beneath, I apply whole numbers for simplicity, just y'all tin have decimal places.

In the dataset with the odd number of observations, notice how the number 12 has six values above information technology and 6 below information technology. Therefore, 12 is the median of this dataset.

When in that location is an fifty-fifty number of values, you count in to the two innermost values and so take the average. The boilerplate of 27 and 29 is 28. Consequently, 28 is the median of this dataset.

Outliers and skewed data have a smaller effect on the median. To empathize why, imagine we have the Median dataset below and notice that the median is 46. Even so, nosotros discover data entry errors and need to modify iv values, which are shaded in the Median Fixed dataset. We'll make them all significantly college so that we now have a skewed distribution with large outliers.

Every bit you can run across, the median doesn't change at all. It is notwithstanding 46. Dissimilar the mean, the median value doesn't depend on all the values in the dataset. Consequently, when some of the values are more farthermost, the consequence on the median is smaller. Of course, with other types of changes, the median can alter. When you lot have a skewed distribution, the median is a better measure of central tendency than the mean.

Related post: Skewed Distributions

Comparing the mean and median

Now, let'southward exam the median on the symmetrical and skewed distributions to see how it performs, and I'll include the mean on the histograms then we tin make comparisons.

In a symmetric distribution, the mean and median both find the center accurately. They are approximately equal.

In a skewed distribution, the outliers in the tail pull the mean away from the center towards the longer tail. For this example, the mean and median differ by over 9000, and the median better represents the central tendency for the distribution.

These data are based on the U.Southward. household income for 2006. Income is the classic example of when to use the median because it tends to be skewed. The median indicates that half of all incomes autumn below 27581, and half are above it. For these data, the mean overestimates where most household incomes fall.

To learn more almost incomes and their right-skewed distributions, read my post about Global Income Distributions.

Statisticians say that the median is a robust statistical while the hateful is sensitive to outliers and skewed distributions.

When to apply the median: Skewed distribution, Continuous information, Ordinal data

Related posts: Median Definition and Uses and What are Robust Statistics?

Mode

The way is the value that occurs the virtually frequently in your information set. On a bar nautical chart, the mode is the highest bar. If the data have multiple values that are tied for occurring the nigh often, you have a multimodal distribution. If no value repeats, the data do not have a mode.

In the dataset below, the value five occurs well-nigh oftentimes, which makes it the style. These data might represent a five-point Likert scale.

Typically, you use the mode with categorical, ordinal, and discrete information. In fact, the mode is the only measure of central tendency that you can use with categorical data—such as the near preferred flavor of water ice cream. However, with categorical data, there isn't a primal value because you lot can't order the groups. With ordinal and discrete data, the way tin can be a value that is not in the center. Again, the mode represents the about common value.

In the graph of service quality, Very Satisfied is the mode of this distribution because it is the nigh common value in the data. Notice how information technology is at the farthermost terminate of the distribution. I'yard sure the service providers are pleased with these results!

Related post: Bar Charts: Using, Examples, and Interpreting

Finding the fashion for continuous data

In the continuous information below, no values repeat, which means there is no mode. With continuous data, it is unlikely that two or more values will exist exactly equal because at that place are an infinite number of values between whatever two values.

When you are working with the raw continuous data, don't exist surprised if in that location is no mode. However, you can find the mode for continuous data by locating the maximum value on a probability distribution plot. If you tin place a probability distribution that fits your data, find the elevation value and utilise it equally the style.

The probability distribution plot displays a lognormal distribution that has a mode of 16700. This distribution corresponds to the U.S. household income instance in the median section.

When to use the mode: Chiselled data, Ordinal data, Count data, Probability Distributions

Which is Best—the Hateful, Median, or Style?

When you lot take a symmetrical distribution for continuous information, the mean, median, and mode are equal. In this case, analysts tend to employ the mean because information technology includes all of the data in the calculations. However, if you take a skewed distribution, the median is oftentimes the best measure of central tendency.

When you lot have ordinal data, the median or fashion is usually the best choice. For chiselled information, you have to use the way.

In cases where you are deciding between the hateful and median as the better mensurate of central tendency, you are also determining which types of statistical hypothesis tests are appropriate for your information—if that is your ultimate goal. I have written an article that discusses when to employ parametric (mean) and nonparametric (median) hypothesis tests forth with the advantages and disadvantages of each blazon.

Analysts frequently use measures of fundamental tendency to draw their datasets. Larn how to Clarify Descriptive Statistics in Excel.

If you lot're learning well-nigh statistics and similar the approach I employ in my weblog, check out my Introduction to Statistics eBook!