So, why have we called it a sample mean? This is because, in statistics, samples and populations have very different meanings and these differences are very important, even if, in the case of the mean, they are calculated in the same way.
The mean is essentially a model of your data set. It is the value that is most common. You will notice, however, that the mean is not often one of the actual values that you have observed in your data set. However, one of its important properties is that it minimises error in the prediction of any one value in your data set. That is, it is the value that produces the lowest amount of error from all other values in the data set.
An important property of the mean is that it includes every value in your data set as part of the calculation. In addition, the mean is the only measure of central tendency where the sum of the deviations of each value from the mean is always zero.
The mean has one main disadvantage: it is particularly susceptible to the influence of outliers. These are values that are unusual compared to the rest of the data set by being especially small or large in numerical value. For example, consider the wages of staff at a factory below:. Staff 1 2 3 4 5 6 7 8 9 10 Salary 15k 18k 16k 14k 15k 15k 12k 17k 90k 95k.
The mean is being skewed by the two large salaries. Therefore, in this situation, we would like to have a better measure of central tendency. As we will find out later, taking the median would be a better measure of central tendency in this situation. Another time when we usually prefer the median over the mean or mode is when our data is skewed i. If we consider the normal distribution - as this is the most frequently assessed in statistics - when the data is perfectly normal, the mean, median and mode are identical.
Moreover, they all represent the most typical value in the data set. However, as the data becomes skewed the mean loses its ability to provide the best central location for the data because the skewed data is dragging it away from the typical value. However, the median best retains this position and is not as strongly influenced by the skewed values. This is explained in more detail in the skewed distribution section later in this guide. The median is the middle score for a set of data that has been arranged in order of magnitude.
The median is less affected by outliers and skewed data. In order to calculate the median, suppose we have the data below:. Our median mark is the middle mark - in this case, 56 highlighted in bold. It is the middle mark because there are 5 scores before it and 5 scores after it. This works fine when you have an odd number of scores, but what happens when you have an even number of scores? What if you had only 10 scores?
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A distribution is said to be negatively skewed when the tail on the left side of the histogram is longer than the right side. A key feature of the skewed distribution is that the mean and median have different values and do not all lie at the center of the curve.
There can also be more than one mode in a skewed distribution. Distributions with two or more modes are known as bi-modal or multimodal, respectively. The distribution shape of the data in is bi-modal because there are two modes two values that occur more frequently than any other for the data item variable. Bi-modal Distribution : Some skewed distributions have two or more modes.
The root-mean-square, also known as the quadratic mean, is a statistical measure of the magnitude of a varying quantity, or set of numbers. It can be calculated for a series of discrete values or for a continuously varying function. Its name comes from its definition as the square root of the mean of the squares of the values. This measure is especially useful when a data set includes both positive and negative numbers.
One possible method of assigning an average to this set would be to simply erase all of the negative signs. This would lead us to compute an average of 5.
However, using the RMS method, we would square every number making them all positive and take the square root of the average. Explicitly, the process is to:. The root-mean-square is always greater than or equal to the average of the unsigned values.
Mathematical Means : This is a geometrical representation of common mathematical means. Depending on the characteristic distribution of a data set, the mean, median or mode may be the more appropriate metric for understanding. Assess various situations and determine whether the mean, median, or mode would be the appropriate measure of central tendency. The mode is the value that appears most often in a set of data. Like the statistical mean and median, the mode is a way of expressing, in a single number, important information about a random variable or a population.
The mode is not necessarily unique, since the same maximum frequency may be attained at different values. The most extreme case occurs in uniform distributions, where all values occur equally frequently. For a sample from a continuous distribution, the concept is unusable in its raw form. No two values will be exactly the same, so each value will occur precisely once.
In order to estimate the mode, the usual practice is to discretize the data by assigning frequency values to intervals of equal distance, as with making a histogram, effectively replacing the values with the midpoints of the intervals they are assigned to. The mode is then the value where the histogram reaches its peak. The median is the numerical value separating the higher half of a data sample, a population, or a probability distribution, from the lower half.
The median of a finite list of numbers can be found by arranging all the observations from lowest value to highest value and picking the middle one e. If there is an even number of observations, then there is no single middle value. In this case, the median is usually defined to be the mean of the two middle values. The median can be used as a measure of location when a distribution is skewed, when end-values are not known, or when one requires reduced importance to be attached to outliers e.
For samples, if it is known that they are drawn from a symmetric distribution, the sample mean can be used as an estimate of the population mode. If elements in a sample data set increase arithmetically, when placed in some order, then the median and arithmetic mean are equal.
The mean is 2. In this case, the arithmetic mean is 6. In general the average value can vary significantly from most values in the sample, and can be larger or smaller than most of them. While the arithmetic mean is often used to report central tendencies, it is not a robust statistic, meaning that it is greatly influenced by outliers values that are very much larger or smaller than most of the values.
Robust statistics are statistics with good performance for data drawn from a wide range of probability distributions, especially for distributions that are not normally distributed. One motivation is to produce statistical methods that are not unduly affected by outliers.
Another motivation is to provide methods with good performance when there are small departures from parametric distributions. Unlike median, the concept of mean makes sense for any random variable assuming values from a vector space. For example, a distribution of points in the plane will typically have a mean and a mode, but the concept of median does not apply. In any voting system where a plurality determines victory, a single modal value determines the victor, while a multi-modal outcome would require some tie-breaking procedure to take place.
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