A list of numbers, or numerical data, can be described by various **statistical measures**. To evaluate data correctly, you need to know the **central tendency of numbers** and the **dispersion** of their values.

A measurement of central tendency is a value that’s typical, or representative, of a group of numbers or other information. Common tools for describing a central tendency include average (arithmetic mean), median, mode, and weighted mean. Dispersion tells you how spread out the values are from the center. If dispersion is small, the values are clustered around the mean. But a wide dispersion of values tells you that the mean average isn’t a reliable representative of all the values.

The mean is simply the average value of a set of numbers. It is also called the average (or arithmetic mean). To find the mean, **sum the values of all the elements and divide by the number of elements**. This also indicates sum of items equals the average multiplied by number of items.

For example, the mean of the set {99, 100, 100, 105, 106, 116, 123} is sum of elements 99 + 100 + 100 + 105 + 106 + 116 + 123 = 749 divided by 7. Thus, the mean is 107.

The median is the **middle value** in a set. By the middle value, we mean the value that has just as many elements in the set that are less than it as it has elements in the set that are greater than it.

For example, the median of the set {99, 100, 100, 105, 106, 116, 123} is 105 because it is the middle number in a set of seven. If you have an **odd number of values**, just select the middle value.

If there is an **even number of elements** in the set, then the median is the number halfway between the two middle numbers. For example, in the set {105, 108, 110, 112, 114, 115}, the median value is 111, since this is halfway between the two middle values 110 and 112.

The mode is the **most common element** in a set. The mode is the value that occurs most frequently in a set of values.

For example, in the set {99, 100, 100, 105, 106, 116, 123}, the mode is 100, because this number appears twice (maximum number of times), while all other elements appear only once.

You determine a weighted mean when some values in a set contribute more to the final average than others. Multiply each individual value by the number of times it occurs in a set of numbers. Then, you add these products together and divide the sum by the total number of times all the values occur.

The easiest measure of dispersion to calculate is the range. The range of a set is the difference between the largest and the smallest elements. The range of values in statistics can come from either a population or a sample. The population is the set of all objects or things, that is, the total amount of all data considered. A sample is just a part of the population.

Though the range is a measure of spread from the highest to the lowest value, it doesn't take the numbers in between.

For example, the range of the set {105, 106, 107, 108, 109, 110, 111, 112, 113} is 8, since 113 – 105 = 8.

The standard deviation expresses variation by measuring **how spread out the distribution is from the mean**. Although the range can give you an idea of the total spread, standard deviation is a more reliable indicator of dispersion because it considers all the data, not just the two on each end.

For example, suppose you get a grade of 75 on a test where the mean grade is 70 and the vast majority of all the other grades fall between 60 and 80. Your score is comparatively better in this situation than if you get a 75 on the same test, where the mean grade is still 70, but most of the grades fall between 45 and 95. In the first situation, the grades are more tightly clustered around the central tendency. A standard deviation in this case is a small number. Your grade is higher compared to all the other test-takers’ grades in the first group than your grade would be in the second scenario. In the second scenario, the standard deviation is a bigger number, and a grade of 75 isn’t as good relative to the others.

A small value for the standard deviation means that the values of the group are more tightly clustered around the mean. A greater standard deviation means that the numbers are more scattered away from the mean. The greater the standard deviation for a group of values, the easier deviating from the center is. The smaller the standard deviation, the harder it is to deviate from the center.