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Measures of Primal Trend

Introduction

A measure of fundamental tendency is a single value that attempts to describe a ready of information by identifying the cardinal position within that set of data. As such, measures of central tendency are sometimes chosen measures of central location. They are also classed as summary statistics. The mean (often called the average) is most likely the measure of key tendency that y'all are most familiar with, only there are others, such as the median and the mode.

The mean, median and mode are all valid measures of central tendency, but under different atmospheric condition, some measures of central tendency become more than advisable to employ than others. In the post-obit sections, we will look at the mean, mode and median, and learn how to calculate them and under what weather they are most appropriate to be used.

Mean (Arithmetics)

The hateful (or average) is the nigh popular and well known measure out of central tendency. It can be used with both discrete and continuous data, although its use is almost oft with continuous data (see our Types of Variable guide for information types). The hateful is equal to the sum of all the values in the data set divided by the number of values in the data set. So, if nosotros take $ n $ values in a data set and they have values $ x_1, x_2, $ …$, x_n $, the sample mean, usually denoted by $ \overline{x} $ (pronounced "x bar"), is:

$$ \overline{10} = {{x_1 + x_2 + \dots + x_n}\over{due north}} $$

This formula is normally written in a slightly different mode using the Greek capitol letter, $ \sum $, pronounced "sigma", which ways "sum of...":

$$ \overline{x} = {{\sum{ten}}\over{northward}} $$

Y'all may have noticed that the higher up formula refers to the sample mean. And so, why take we called it a sample hateful? This is because, in statistics, samples and populations have very different meanings and these differences are very important, fifty-fifty if, in the case of the mean, they are calculated in the same fashion. To acknowledge that we are calculating the population mean and non the sample mean, we use the Greek lower case letter "mu", denoted every bit $ \mu $:

$$ \mu = {{\sum{x}}\over{due north}} $$

The mean is essentially a model of your data set. It is the value that is well-nigh mutual. You will notice, however, that the mean is non often ane of the bodily values that you lot have observed in your data set. Even so, one of its important backdrop is that it minimises mistake in the prediction of any one value in your data set. That is, it is the value that produces the lowest amount of error from all other values in the data set.

An important property of the mean is that information technology includes every value in your data set as office of the adding. In addition, the mean is the only mensurate of central trend where the sum of the deviations of each value from the mean is e'er zero.

When not to use the mean

The mean has 1 main disadvantage: it is particularly susceptible to the influence of outliers. These are values that are unusual compared to the residual of the information set by being especially minor or large in numerical value. For case, consider the wages of staff at a mill below:

Staff	1	two	iii	4	5	6	seven	8	9	10
Salary	15k	18k	16k	14k	15k	15k	12k	17k	90k	95k

The mean salary for these ten staff is $30.7k. However, inspecting the raw data suggests that this hateful value might non be the best fashion to accurately reflect the typical salary of a worker, as near workers take salaries in the $12k to 18k range. The mean is existence skewed by the 2 big salaries. Therefore, in this state of affairs, we would like to have a better measure out of central tendency. As we will detect out after, taking the median would exist a better measure out of cardinal tendency in this situation.

Another time when we ordinarily adopt the median over the mean (or mode) is when our data is skewed (i.e., the frequency distribution for our information is skewed). If we consider the normal distribution - as this is the virtually frequently assessed in statistics - when the data is perfectly normal, the mean, median and mode are identical. Moreover, they all stand for the most typical value in the data set. Nonetheless, as the data becomes skewed the mean loses its ability to provide the best key location for the data because the skewed data is dragging it away from the typical value. However, the median best retains this position and is non every bit strongly influenced past the skewed values. This is explained in more detail in the skewed distribution section afterwards in this guide.

Median

The median is the centre score for a set of information that has been arranged in society of magnitude. The median is less affected past outliers and skewed data. In order to calculate the median, suppose nosotros take the data below:

Nosotros first need to rearrange that data into order of magnitude (smallest first):

fourteen

Our median marking is the middle marker - in this instance, 56 (highlighted in assuming). Information technology is the middle marker considering there are 5 scores before it and five scores after it. This works fine when you take an odd number of scores, just what happens when yous have an even number of scores? What if you lot had only 10 scores? Well, you simply have to take the centre two scores and average the event. Then, if nosotros look at the example below:

We over again rearrange that data into gild of magnitude (smallest offset):

fourteen

Only now we have to take the 5th and 6th score in our information gear up and average them to get a median of 55.5.

Style

The mode is the nearly frequent score in our data set. On a histogram it represents the highest bar in a bar chart or histogram. You can, therefore, sometimes consider the mode every bit being the most popular option. An example of a mode is presented below:

Histogram showing mode as highest bar in the middle of the continuous distribution as the mode

Ordinarily, the mode is used for categorical data where nosotros wish to know which is the nigh common category, as illustrated below:

Bar chart showing highest bar as the mode

We tin run across in a higher place that the near common form of ship, in this particular information ready, is the passenger vehicle. However, i of the problems with the way is that information technology is not unique, so information technology leaves us with problems when we accept ii or more than values that share the highest frequency, such every bit below:

Histogram of a continuous distribution showing two modes, both somewhat centrally located

We are at present stuck as to which mode best describes the key tendency of the data. This is particularly problematic when we have continuous data because we are more likely not to accept any ane value that is more frequent than the other. For example, consider measuring 30 peoples' weight (to the nearest 0.1 kg). How probable is information technology that nosotros volition find two or more than people with exactly the same weight (e.thou., 67.4 kg)? The reply, is probably very unlikely - many people might be close, but with such a small sample (xxx people) and a large range of possible weights, you are unlikely to detect two people with exactly the same weight; that is, to the nearest 0.one kg. This is why the manner is very rarely used with continuous data.

Another trouble with the style is that information technology volition not provide u.s.a. with a very expert measure of key trend when the near common mark is far away from the rest of the data in the data set, as depicted in the diagram beneath:

Histogram of a continuous distribution showing mode not centrally located

In the in a higher place diagram the mode has a value of 2. Nosotros can clearly see, nonetheless, that the fashion is not representative of the data, which is mostly concentrated around the 20 to thirty value range. To use the mode to draw the central tendency of this data set would exist misleading.

Skewed Distributions and the Mean and Median

We often test whether our information is normally distributed because this is a common assumption underlying many statistical tests. An case of a usually distributed fix of information is presented below:

A histogram showing a normally distributed continuous data set

When you have a normally distributed sample you lot can legitimately use both the mean or the median every bit your measure of central trend. In fact, in any symmetrical distribution the mean, median and fashion are equal. However, in this situation, the mean is widely preferred as the all-time measure out of central trend because it is the measure out that includes all the values in the data ready for its calculation, and whatsoever change in any of the scores will affect the value of the mean. This is not the case with the median or mode.

Yet, when our information is skewed, for example, equally with the right-skewed information set below:

Histogram of a skewed distribution showing a noticable difference between the median and mean values

We notice that the hateful is beingness dragged in the directly of the skew. In these situations, the median is more often than not considered to exist the best representative of the central location of the data. The more than skewed the distribution, the greater the difference between the median and mean, and the greater emphasis should be placed on using the median as opposed to the mean. A classic example of the higher up right-skewed distribution is income (salary), where higher-earners provide a false representation of the typical income if expressed as a hateful and non a median.

If dealing with a normal distribution, and tests of normality show that the data is non-normal, it is customary to use the median instead of the mean. Even so, this is more than a rule of pollex than a strict guideline. Sometimes, researchers wish to report the mean of a skewed distribution if the median and hateful are not appreciably different (a subjective assessment), and if it allows easier comparisons to previous research to be fabricated.