The mean often called the average is most likely the measure of central tendency that you are most familiar with, but 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 conditions, some measures of central tendency become more appropriate to use than others.
In the following sections, we will look at the mean, mode and median, and learn how to calculate them and under what conditions they are most appropriate to be used. The mean or average is the most popular and well known measure of central tendency.
It can be used with both discrete and continuous data, although its use is most often with continuous data see our Types of Variable guide for data types. The mean is equal to the sum of all the values in the data set divided by the number of values in the data set. You may have noticed that the above formula refers to the sample mean.
So, why have we called it a sample mean? This is because, in statistics, samples and populations have very different meanings and these differences are very important, even if, in the case of the mean, they are calculated in the same way.
The mean is essentially a model of your data set. It is the value that is most common. You will notice, however, that the mean is not often one of the actual values that you have observed in your data set. However, one of its important properties is that it minimises error 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 it includes every value in your data set as part of the calculation. In addition, the mean is the only measure of central tendency where the sum of the deviations of each value from the mean is always zero. The mean has one main disadvantage: it is particularly susceptible to the influence of outliers.
These are values that are unusual compared to the rest of the data set by being especially small or large in numerical value. For example, consider the wages of staff at a factory below:. Staff 1 2 3 4 5 6 7 8 9 10 Salary 15k 18k 16k 14k 15k 15k 12k 17k 90k 95k. The mean is being skewed by the two large salaries. Therefore, in this situation, we would like to have a better measure of central tendency. As we will find out later, taking the median would be a better measure of central tendency in this situation.
Another time when we usually prefer the median over the mean or mode is when our data is skewed i. If we consider the normal distribution - as this is the most frequently assessed in statistics - when the data is perfectly normal, the mean, median and mode are identical. Moreover, they all represent the most typical value in the data set. However, as the data becomes skewed the mean loses its ability to provide the best central 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 not as strongly influenced by the skewed values. This is explained in more detail in the skewed distribution section later in this guide. In this way, it calculates a number the t-value illustrating the magnitude of the difference between the two group means being compared, and estimates the likelihood that this difference exists purely by chance p-value. Your choice of t-test depends on whether you are studying one group or two groups, and whether you care about the direction of the difference in group means.
If you are studying one group, use a paired t-test to compare the group mean over time or after an intervention, or use a one-sample t-test to compare the group mean to a standard value. If you are studying two groups, use a two-sample t-test. If you want to know only whether a difference exists, use a two-tailed test.
If you want to know if one group mean is greater or less than the other, use a left-tailed or right-tailed one-tailed test. A t-test is a statistical test that compares the means of two samples. It is used in hypothesis testing , with a null hypothesis that the difference in group means is zero and an alternate hypothesis that the difference in group means is different from zero.
Statistical significance is a term used by researchers to state that it is unlikely their observations could have occurred under the null hypothesis of a statistical test. Significance is usually denoted by a p -value , or probability value. Statistical significance is arbitrary — it depends on the threshold, or alpha value, chosen by the researcher. When the p -value falls below the chosen alpha value, then we say the result of the test is statistically significant.
A test statistic is a number calculated by a statistical test. It describes how far your observed data is from the null hypothesis of no relationship between variables or no difference among sample groups. The test statistic tells you how different two or more groups are from the overall population mean , or how different a linear slope is from the slope predicted by a null hypothesis.
Different test statistics are used in different statistical tests. The measures of central tendency you can use depends on the level of measurement of your data. Ordinal data has two characteristics:. Nominal and ordinal are two of the four levels of measurement.
Nominal level data can only be classified, while ordinal level data can be classified and ordered. If your confidence interval for a difference between groups includes zero, that means that if you run your experiment again you have a good chance of finding no difference between groups.
If your confidence interval for a correlation or regression includes zero, that means that if you run your experiment again there is a good chance of finding no correlation in your data. In both of these cases, you will also find a high p -value when you run your statistical test, meaning that your results could have occurred under the null hypothesis of no relationship between variables or no difference between groups.
If you want to calculate a confidence interval around the mean of data that is not normally distributed , you have two choices:. The standard normal distribution , also called the z -distribution, is a special normal distribution where the mean is 0 and the standard deviation is 1.
Any normal distribution can be converted into the standard normal distribution by turning the individual values into z -scores. In a z -distribution, z -scores tell you how many standard deviations away from the mean each value lies. The z -score and t -score aka z -value and t -value show how many standard deviations away from the mean of the distribution you are, assuming your data follow a z -distribution or a t -distribution.
These scores are used in statistical tests to show how far from the mean of the predicted distribution your statistical estimate is. If your test produces a z -score of 2. The predicted mean and distribution of your estimate are generated by the null hypothesis of the statistical test you are using. The more standard deviations away from the predicted mean your estimate is, the less likely it is that the estimate could have occurred under the null hypothesis.
To calculate the confidence interval , you need to know:. Then you can plug these components into the confidence interval formula that corresponds to your data. The formula depends on the type of estimate e. The confidence level is the percentage of times you expect to get close to the same estimate if you run your experiment again or resample the population in the same way.
The confidence interval is the actual upper and lower bounds of the estimate you expect to find at a given level of confidence. These are the upper and lower bounds of the confidence interval. Nominal data is data that can be labelled or classified into mutually exclusive categories within a variable. These categories cannot be ordered in a meaningful way. For example, for the nominal variable of preferred mode of transportation, you may have the categories of car, bus, train, tram or bicycle.
Statistical tests commonly assume that:. If your data does not meet these assumptions you might still be able to use a nonparametric statistical test , which have fewer requirements but also make weaker inferences. Measures of central tendency help you find the middle, or the average, of a data set. Some variables have fixed levels.
For example, gender and ethnicity are always nominal level data because they cannot be ranked. However, for other variables, you can choose the level of measurement. For example, income is a variable that can be recorded on an ordinal or a ratio scale:. If you have a choice, the ratio level is always preferable because you can analyze data in more ways. The higher the level of measurement, the more precise your data is.
The level at which you measure a variable determines how you can analyze your data. Depending on the level of measurement , you can perform different descriptive statistics to get an overall summary of your data and inferential statistics to see if your results support or refute your hypothesis. Levels of measurement tell you how precisely variables are recorded. There are 4 levels of measurement, which can be ranked from low to high:.
The p -value only tells you how likely the data you have observed is to have occurred under the null hypothesis. The alpha value, or the threshold for statistical significance , is arbitrary — which value you use depends on your field of study. In most cases, researchers use an alpha of 0. P -values are usually automatically calculated by the program you use to perform your statistical test. They can also be estimated using p -value tables for the relevant test statistic. P -values are calculated from the null distribution of the test statistic.
They tell you how often a test statistic is expected to occur under the null hypothesis of the statistical test, based on where it falls in the null distribution. If the test statistic is far from the mean of the null distribution, then the p -value will be small, showing that the test statistic is not likely to have occurred under the null hypothesis. A p -value , or probability value, is a number describing how likely it is that your data would have occurred under the null hypothesis of your statistical test.
You can choose the right statistical test by looking at what type of data you have collected and what type of relationship you want to test. The test statistic will change based on the number of observations in your data, how variable your observations are, and how strong the underlying patterns in the data are. For example, if one data set has higher variability while another has lower variability, the first data set will produce a test statistic closer to the null hypothesis, even if the true correlation between two variables is the same in either data set.
Want to contact us directly? No problem. We are always here for you. Scribbr specializes in editing study-related documents. We proofread:. You can find all the citation styles and locales used in the Scribbr Citation Generator in our publicly accessible repository on Github. Frequently asked questions See all. Frequently asked questions: Statistics What does standard deviation tell you? How do I find the median?
Can there be more than one mode? Your data can be: without any mode unimodal, with one mode, bimodal, with two modes, trimodal, with three modes, or multimodal, with four or more modes. How do I find the mode? To find the mode : If your data is numerical or quantitative, order the values from low to high. If it is categorical, sort the values by group, in any order. Then you simply need to identify the most frequently occurring value. When should I use the interquartile range? What are the two main methods for calculating interquartile range?
What is homoscedasticity? What is variance used for in statistics? Both measures reflect variability in a distribution, but their units differ: Standard deviation is expressed in the same units as the original values e. Variance is expressed in much larger units e. What is the empirical rule? Around What is a normal distribution? When should I use the median? Can the range be a negative number?
What is the range in statistics? What are the 4 main measures of variability? Variability is most commonly measured with the following descriptive statistics : Range : the difference between the highest and lowest values Interquartile range : the range of the middle half of a distribution Standard deviation : average distance from the mean Variance : average of squared distances from the mean.
What is variability? Variability is also referred to as spread, scatter or dispersion. What is the difference between interval and ratio data? What is a critical value? What is the difference between the t-distribution and the standard normal distribution? What is a t-score? What is a t-distribution?
Is the correlation coefficient the same as the slope of the line? What do the sign and value of the correlation coefficient tell you? What are the assumptions of the Pearson correlation coefficient? What is a correlation coefficient? How do you increase statistical power? There are various ways to improve power: Increase the potential effect size by manipulating your independent variable more strongly, Increase sample size, Increase the significance level alpha , Reduce measurement error by increasing the precision and accuracy of your measurement devices and procedures, Use a one-tailed test instead of a two-tailed test for t tests and z tests.
What is a power analysis? Sample size : the minimum number of observations needed to observe an effect of a certain size with a given power level.
Expected effect size : a standardized way of expressing the magnitude of the expected result of your study, usually based on similar studies or a pilot study.
The mode is the least used of the measures of central tendency and can only be used when dealing with nominal data. For this reason, the mode will be the best measure of central tendency as it is the only one appropriate to use when dealing with nominal data. The median is usually preferred to other measures of central tendency when your data set is skewed i.
However, the mode can also be appropriate in these situations, but is not as commonly used as the median. The median is usually preferred in these situations because the value of the mean can be distorted by the outliers.
However, it will depend on how influential the outliers are. If they do not significantly distort the mean, using the mean as the measure of central tendency will usually be preferred.
If the data set is perfectly normal, the mean, median and mean are equal to each other i. The median and mean can only have one value for a given data set. The mode can have more than one value see Mode section on previous page. FAQs - Measures of Central Tendency Please find below some common questions that are asked regarding measures of central tendency, along with their answers.
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