Central Measures Measures of central tendencies are nothing but the measures to describe the “central” values of a collected sample. For an ungrouped set of data these measures are: the mean, the median, and the mode. The Mean The arithmetic mean, or the simple mean, is computed by summing all numbers in an array of numbers (x’) and then dividing by the number of observations (N) in the array. ; the sum is over all i’s.
The mean uses all of the observations, and each observation affects the mean.
Even though the mean is sensitive to extreme values – that is extremely large or small data an cause the mean to be pulled toward the extreme data – it is still the most widely used measure of location. This is due to the fact that the mean has valuable mathematical properties that make it convenient for use with inferential statistical analysis. For example: the sum of the deviations of the numbers in a set of data from the mean is zero; and the sum of the squared deviations of the numbers in a set of data from the mean is the minimum value.
Central Measures analysis For example: the minimum value. The Median The median is the middle value in an ordered array of observations.
If there is an ven number of observations in the array, the median is the average of the two middle numbers. If there is an odd number of data in the array, the median is the middle number. The median is often used to summarize the distribution of an outcome. If the distribution is skewed, the median and the interquartile range may be better than other measures to indicate where the observed data are concentrated.
Generally, the median provides a better measure of location than the mean when there are some extremely large or small observations, that is, when the data are skewed to the right or to the left.
Note that if the median is less than the mean, the data set is skewed to the right. If the median is greater than the mean, the data set is skewed to the left. The mean has two distinct advantages over the median. the two means. The Mode The mode is the most frequently occurring value in a set of observations.
Why use the mode? The classic example is the shirt/shoe manufacturer who wants to decide what sizes to introduce.
The manufacturing company may have to accept two sets of sizes: one set for men; and one set for women. In this case we say the data are bimodal. Sets of observations with more than two modes are referred to as multimodal. Note that the mode is not a helpful measure of location, because there can be more than one mode or even no mode. When the mean and the median are known, it is possible to estimate the mode for the unimodal distribution using the other two averages as follows: Mode 3*(median)2*(mean) This estimate is applicable to both grouped and ungrouped data sets.
Whenever, more than one mode exist, then the population from which the sample came is a mixture of more than one population. However, note that a Uniform distribution has ncountable number of modes having equal density value; therefore it is considered as a homogeneous population. Standard Error of Mean Once we calculated one of the measures of the central tendency, say the sample mean, one might like to know how good is our estimated sample mean? We answer this by calculating a confidence interval for the sample mean.
For this purpose we need to calculate the standard error of the mean (SEM). The standard error is the error in estimating the population mean using a sample drawn from that population. By definition the standard error of the sample mean is nothing but the standard eviation of the sample mean.
What it means in theory is that we will have to repeat our sampling procedure several times to make, say, m samples. Then calculate the sample mean for each sample, resulting in m sample means.
The standard deviation of the m sample means about the super mean (the mean of all sample means) is known as the standard error of the sample mean (SEM). However, in practice SEM can be calculated from the sample standard deviation (S) as: SEM = S / sqrt(N) where N – number of observations; and Confidence Interval for Mean A standard deviation or a standard error has little practical use in itself. But it becomes meaningful when we use it to calculate confidence intervals.