# statistical mean, median, mode and range

The terms mean, median and mode are used to describe the central tendency of a large data set. Range provides provides context for the mean, median and mode.

When working with a large data set, it can be useful to represent the entire data set with a single value that describes the "middle" or "average" value of the entire set. In statistics, that single value is called the central tendency and mean, median and mode are all ways to describe it. To find the mean, add up the values in the data set and then divide by the number of values that you added. To find the median, list the values of the data set in numerical order and identify which value appears in the middle of the list. To find the mode, identify which value in the data set occurs most often. Range, which is the difference between the largest and smallest value in the data set, describes how well the central tendency represents the data. If the range is large, the central tendency is not as representative of the data as it would be if the range was small.

### How are mean, median, mode and range used in the data center?

IT professionals need to understand the definition of mean, median, mode and range to plan capacity and balance load, manage systems, perform maintenance and troubleshoot issues. These various tasks dictate that the administrator calculate mean, median, mode or range, or often some combination, to show a statistically significant quantity, trend or deviation from the norm. Finding the mean, median, mode and range is only the start. The administrator then needs to apply this information to investigate root causes of a problem, accurately forecast future needs or set acceptable working parameters for IT systems.

### Mean

The mean is the average of all numbers and is sometimes called the arithmetic mean. To calculate mean, add together all of the numbers in a set and then divide the sum by the total count of numbers. For example, in a data center rack, five servers consume 100 watts, 98 watts, 105 watts, 90 watts and 102 watts of power, respectively. The mean power use of that rack is calculated as (100 + 98 + 105 + 90 + 102 W)/5 servers = a calculated mean of 99 W per server. Intelligent power distribution units report the mean power utilization of the rack to systems management software.

### Median

In the data center, means and medians are often tracked over time to spot trends, which inform capacity planning or power cost predictions.The statistical median is the middle number in a sequence of numbers. To find the median, organize each number in order by size; the number in the middle is the median. For the five servers in the rack, arrange the power consumption figures from lowest to highest: 90 W, 98 W, 100 W, 102 W and 105 W. The median power consumption of the rack is 100 W. If there is an even set of numbers, average the two middle numbers. For example, if the rack had a sixth server that used 110 W, the new number set would be 90 W, 98 W, 100 W, 102 W, 105 W and 110 W. Find the median by averaging the two middle numbers: (100 + 102)/2 = 101 W.

### Mode

The mode is the number that occurs most often within a set of numbers. For the server power consumption examples above, there is no mode because each element is different. But suppose the administrator measured the power consumption of an entire netowork operations center (NOC) and the set of numbers is 90 W, 104 W, 98 W, 98 W, 105 W, 92 W, 102 W, 100 W, 110 W, 98 W, 210 W and 115 W. The mode is 98 W since that power consumption measurement occurs most often amongst the 12 servers. Mode helps identify the most common or frequent occurrence of a characteristic. It is possible to have two modes (bimodal), three modes (trimodal) or more modes within larger sets of numbers.

### Range

The range is the difference between the highest and lowest values within a set of numbers. To calculate range, subtract the smallest number from the largest number in the set. If a six-server rack includes 90 W, 98 W, 100 W, 102 W, 105 W and 110 W, the power consumption range is 110 W - 90 W = 20 W.

Range shows how much the numbers in a set vary. Many IT systems operate within an acceptable range; a value in excess of that range might trigger a warning or alarm to IT staff. To find the variance in a data set, subtract each number from the mean, and then square the result. Find the average of these squared differences, and that is the variance in the group. In our original group of five servers, the mean was 99. The 100 W-server varies from the mean by 1 W, the 105 W-server by 6 W, and so on. The squares of each difference equal 1, 1, 36, 81 and 9. So to calculate the variance, add 1 + 1 + 36 + 81 + 9 and divide by 5. The variance is 25.6. Standard deviation denotes how far apart all the numbers are in a set. The standard deviation is calculated by finding the square root of the variance. In this example, the standard deviation is 5.1.

Interquartile range, the middle fifty or midspread of a set of numbers, removes the outliers -- highest and lowest numbers in a set. If there is a large set of numbers, divide them evenly into lower and higher numbers. Then find the median of each of these groups. Find the interquartile range by subtracting the lower median from the higher median. If a rack of six servers' power wattage is arranged from lowest to highest: 90, 98, 100, 102, 105, 110, divide this set into low numbers (90, 98, 100) and high numbers (102, 105, 110). Find the median for each: 98 and 105. Subtract the lower median from the higher median: 105 watts - 98 W = 7 W, which is the interquartile range of these servers.

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