Measures Of Dispersion Essay

Lesson 5-3: Measures of Dispersion When comparing two or more sets of data, using a measure of central tendency may not always be adequate. Therefore, it is necessary to study additional measures to more accurately describe the dataset. Dispersion is a general term that refers to the spread of a series of values from a central value or average, such as the median or mean. Measures of Dispersion A measure of dispersion or variability is a method of measuring the degree to which numerical data or values tend to spread from or cluster around a central point or average. The most common measures of dispersion are the range, standard deviation, and variance. The Range The range of a set of ungrouped data is the difference between the greatest and least data values.
It is a very unstable measure as it only depends on two extreme measurements – the lowest and the highest values. Another measure of dispersion that is less sensitive to extreme values is the standard deviation. The standard deviation of a data set makes use of the individual amount that each data value deviates from the mean. The standard deviation is by far considered the most important measure of variability as compared to the other measures. To find the standard deviation of a set of ungrouped data, we perform the following steps (assuming that a sample is given): There are no major writing issues in this paragraph. However, I would suggest adding a title or subtitle to provide context for the reader. Calculate the square root of the quotient obtained in step 5 to determine the value of the standard deviation for the ungrouped data. To compute the standard deviation for a population of N numbers with a mean μ, use the formula σ=√((∑▒(x-μ)^2 )/N). Similarly, for a sample of n numbers with a mean x ̃, use the formula s=√((∑▒(x-x ̅ )^2 )/(n-1)). Please note that titles, subtitles, quotations, and citations should remain unchanged. There are a few issues with the paragraph that need to be fixed: x̅ = 51/5 = 10.2 s = √((∑(x - x̅)^2)/(n - 1)) = √(42.8/4) = 3.27 Standard Deviation for Grouped Data The standard deviation of a frequency distribution can be computed by the following formula: Sample standard deviation: s = √((∑f(m - x̅)^2)/(n - 1)) Population standard deviation: σ = √((∑f(m - μ)^2)/N) where: s = sample standard deviation σ = population standard deviation f = class frequency m = class midpoint