# Define measures of central tendency variability and relationship

### Psych. Statistics: Central Tendency and Variability In statistics, a central tendency is a central or typical value for a probability distribution. It may also be called a center or location of the distribution. Colloquially, measures of central tendency are often called averages. one asks for a measure of central tendency that minimizes variation: such that variation from the center. Central tendency is defined as “the statistical measure that identifies a single [3 ] The standard statistical notations (in relation to measures of central tendency) are As the variability in the data increases, the difference among these means . It balances the distances of observations to the mean. Another measure of central tendency is the median, which is defined as the middle value.

The problem is that there is no single measure that will always produce a central, representative value in every situation. There are three main measures of central tendency: The mean of a set of scores abbreviated M is the most common and useful measure of central tendency. The mean is the sum of the scores divided by the total number of scores.

The mean is commonly known as the arithmetic average. The mean can only be used for variables at the interval or ratio levels of measurement. One can think of the mean as the balance point of a distribution the center of gravity. It balances the distances of observations to the mean. Another measure of central tendency is the median, which is defined as the middle value when the numbers are arranged in increasing or decreasing order.

The median is the score that divides the distribution of scores exactly in half. The median is also the 50th percentile. The median can be used for variables at the ordinal, interval or ratio levels of measurement. For odd number of count the median is middle value. If there is an even number of items in a set, the median is the average of the two middle values. The median may sometimes be a better indicator of central tendency than the mean, especially when there are extreme values. Another indicator of central tendency is the mode, or the value that occurs most often in a set of numbers. In other words, the mode is the score or category of scores in a frequency distribution that has the greatest frequency.

The mode can be used for variables at any level of measurement nominal, ordinal, interval or ratio. Sometimes a distribution has more than one mode. Such a distribution is called multimodal. A distribution with two modes is called bimodal. Note that the modes do not have to have the same frequencies. The tallest peak is called the major mode; other peaks are called minor modes. Some distributions do not have modes. A rectangular distribution has no mode. 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.

### Central tendency - Wikipedia

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

### Measures of Central tendency and Variability | All about Business and management

The median, you recall, falls at the 50th percentile. 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.

Relationship Between Mean, Median & Mode - Class 11 Economics Measures of Central Tendency

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.

## Measures of central tendency: The mean

This is not so much the case for nominal and ordinal variables. If the variable is nominal, obviously the mode is the only measure of central tendency to use. If the variable is ordinal, the median is probably your best bet because it provides more information about the sample than the mode does. If the distribution is symmetrical, the mean is the best measure of central tendency.

If the distribution is skewed either positively or negatively, the median is more accurate. As an example of why the mean might not be the best measure of central tendency for a skewed distribution, consider the following passage from Charles Wheelan's Naked Statistics: