# Measures of central tendency standard deviation and variance relationship

### Measures of Variability Variance (S2) = average squared deviation of values from mean For example, a measure of two large companies with a difference of \$10, in Standard deviation is only used to measure spread or dispersion around. The mean, median and mode are all valid measures of central tendency, but under different You may have noticed that the above formula refers to the sample mean. However, one of its important properties is that it minimises error in the. The variance of the data is the average squared distance between the mean and each data value. σ 2 = ∑ i The formula for the variance is. σ 2 = ∑ i . In statistics, the standard deviation is a very common measure of dispersion. Standard.

Mode The mode is the most frequent score in our data set. On a histogram it represents the highest bar in a bar chart or histogram.

Range, variance and standard deviation as measures of dispersion - Khan Academy

You can, therefore, sometimes consider the mode as being the most popular option. An example of a mode is presented below: Normally, the mode is used for categorical data where we wish to know which is the most common category, as illustrated below: We can see above that the most common form of transport, in this particular data set, is the bus. However, one of the problems with the mode is that it is not unique, so it leaves us with problems when we have two or more values that share the highest frequency, such as below: We are now stuck as to which mode best describes the central tendency of the data. This is particularly problematic when we have continuous data because we are more likely not to have any one value that is more frequent than the other. For example, consider measuring 30 peoples' weight to the nearest 0.

How likely is it that we will find two or more people with exactly the same weight e. The answer, is probably very unlikely - many people might be close, but with such a small sample 30 people and a large range of possible weights, you are unlikely to find two people with exactly the same weight; that is, to the nearest 0. This is why the mode is very rarely used with continuous data.

## Central Tendency & Variability

Another problem with the mode is that it will not provide us with a very good measure of central tendency when the most common mark is far away from the rest of the data in the data set, as depicted in the diagram below: In the above diagram the mode has a value of 2. Observations that are significantly larger or smaller than the others in a sample can impact some statistical measures in such a way as to make them highly misleading, but the median is immune to them. In other words, it doesn't matter if the biggest number is 20 or 20,; it still only counts as one number. The Mean The mean is what people typically refer to as "the average". The mean takes into account the value of every observation and thus provides the most information of any measure of central tendency. Unlike the median, however, the mean is sensitive to outliers. In other words, one extraordinarily high or low value in your dataset can dramatically raise or lower the mean. The mean, often shown as an x or a y variable with a line over it pronounced either "x-bar" or "y-bar"is the sum of all the scores divided by the total number of scores. In statistical notation, we would write it out as follows: In that equation, is the mean, X represents the value of each case and N is the total number of cases. The fact that calculating the mean requires addition and division is the very reason it can't be used with either nominal or ordinal variables.

Percentiles A percentile is a number below which a certain percent of the distribution falls.

## Measures of Central Tendency

For example, if you score in the 90th percentile on a test, 90 percent of the students who took the test scored below you. If you score in the 72nd percentile on a test, 72 percent of the students who took the test scored below you. If scored in the 5th percentile on a test, maybe that subject isn't for you. The median, you recall, falls at the 50th percentile.

### Central Tendency & Variability - Department of Sociology - The University of Utah

Fifty percent of the observations fall below it. Skewed Distributions A symmetrical distribution is a distribution where the mean, median and mode are the same. A skewed distribution, on the other hand, is a distribution with extreme values on one side or the other that force the median away from the mean in one direction or the other.

If the mean is greater than the median, the distribution is said to be positively skewed.

### Measures of central tendency: The mean

In other words, there is an extremely large value that is "pulling" the mean toward the upper end of the distribution. If the mean is smaller than the median, the distribution is said to be negatively skewed. In other words, there is an extremely small value that is "pulling" the mean toward the lower end of the distribution.

Distributions of income are usually positively skewed thanks to the small number of people who make ungodly amounts of money. Consider the admittedly dated case of Major League Soccer players as an extreme example. When trying to decide which measure of central tendency to use, you must consider both level of measurement and skew.

This is not so much the case for nominal and ordinal variables.