Which measure of variability is normally reported with the mean

Percentile: a score below which a specific percentage of a given distribution falls. Positively skewed distribution: a distribution with a handful of extremely large values. Negatively skewed distribution: a distribution with a handful of extremely low values. Measures of variability: numbers that describe the diversity or dispersion in the distribution of a given variable.

Box plot: a graphic representation of the range, interquartile range and median of a given variable. The mode is the category with the greatest frequency or percentage. It is not the frequency itself. In other words, if someone asks you for the mode of the distribution shown below, the answer would be coconut, NOT It is possible to have more than one mode in a distribution.

Such distributions are considered bimodal if there are two modes or multi-modal if there are more than two modes. Distributions without a clear mode are said to be uniform. The mode is not particularly useful, but it is the only measure of central tendency we can use with nominal variables. You will find out why it is the only appropriate measure for nominal variables as we learn about the median and mean next.

The median is the middlemost number. In other words, it's the number that divides the distribution exactly in half such that half the cases are above the median, and half are below.

Conceptually, finding the median is fairly simple and entails only putting all of your observations in order from least to greatest and then finding whichever number falls in the middle. Note that finding the median requires first ordering all of the observations from least to greatest. This is why the median is not an appropriate measure of central tendency for nominal variables, as nominal variables have no inherent order. In practice, finding the median can be a bit more involved, especially if you have a large number of observations—see your textbook for an explanation of how to find the median in such situations.

Some of you are probably already wondering, "What happens if you have an even number of cases? There won't be a middle number then, right? If your dataset has an even number of cases, the median is the average of the two middlemost numbers.

One of the median's advantages is that it is not sensitive to outliers. An outlier is an observation that lies an abnormal distance from other values in a sample. 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.

Consider the following:. These two distributions have identical medians even though Distribution 2 has a very large outlier, which would end up skewing the mean pretty significantly, as we'll see in just a moment. 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. 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. The median, you recall, falls at the 50th percentile. Fifty percent of the observations fall below it. 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. 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. Common measures of variability include range, variance, and standard deviation. The present entry discusses the value of measures of variability, [Page ] specifically in relation to common measures of central tendency.

It also provides basic information about three common measures of variability, including how to calculate the measures. Although researchers often place focus on measures of central tendency, such as mean, median, and mode, measures of central tendency Show page numbers Download PDF.

Search form icon-arrow-top icon-arrow-top. As mentioned before, a small standard deviation coefficient indicates that scores are close together, whilst a large standard deviation coefficient indicates that scores are far apart. In this example, both sets of data have the same mean, but the standard deviation coefficient is different:. In this example, the scores in Set A are 0. So scores in Set B are more dispersed than scores in Set A. Distributions can be asymmetrical or skewed; that is, the tail of the distribution in the positive direction extends further than the tail in the negative direction, or vice versa.

A distribution with the longer tail extending in the positive direction is said to have a positive skew; it is skewed to the right. A distribution with the longer tail extending to the left is negatively skewed, or skewed to the left :. Distributions also differ in terms of whether the data are peaked or flat.

Distributions with positive kurtosis have a distinct peak near the mean and decline rapidly, whilst distributions with negative kurtosis tend to be more flat:. The normal distribution is the most important and commonly used distribution in statistics. It is also known as the bell curve or Gaussian curve. Even though normal distributions can differ in their means and standard deviation, they share some characteristics related to the distribution of scores:.

Knowing the mean and standard deviation of a normal distribution, we can calculate the values that lie within 1 standard deviation of the mean. For example, if the mean of a normal distribution is 25 years age and the standard distribution is 8 years, then:. Descriptive statistics: measures of variability Variability refers to how spread scores are in a distribution out; that is, it refers to the amount of spread of the scores around the mean.

There are four frequently used measures of the variability of a distribution: range interquartile range variance standard deviation. Measures of variability Range Interquartile range Variance Standard deviation Range The most basic measure of variation is the range, which is the distance from the smallest to the largest value in a distribution.

Variance The variance is the average squared difference of the scores from the mean. To calculate the variance:. See column Squared deviation. Finally, the mean of the squared deviations is calculated. The variance is 1.

This bias is in the direction of underestimating the population value.