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Summary Statistics When analysing practical sets of data, it is useful to be able to define a small number of values that summarise the main features present. We will derive (i) representative values, (ii) measures of spread and (iii) measures of skewness and other characteristics. Representative Values These are sometimes called measures of location or measures of central tendency. 1. Random Value Given a set of data S = { x1, x2, … , xn }, we select a random number, say k, in the range 1 to n and return the value xk. This method of generating a representative value is straightforward, but it suffers from the fact that extreme values can occur and successive values could vary considerably from one another. 2. Arithmetic Mean This is also known as the average. For the set S above the average is x = {x1 + x2 + … + xn }/ n. If x1 occurs f1 times, x2 occurs f2 times and so on, we get the formula x = { f 1 x1 + f 2 x2 + … + f n xn } / { f 1 + f 2 + … + f n } , written x = fx / f , where (sigma) denotes a sum. Example 1. The data refers to the marks that students in a class obtained in an examination. Find the average mark for the class. The first point to note is that the marks are presented as Mark Mid-Point Number ranges, so we must be careful in our of Range of Students interpretation of the ranges. All the intervals xi fi f i xi must be of equal rank and their must be no gaps in the classification. In our case, we 0 - 19 10 2 20 interpret the range 0 - 19 to contain marks 21 - 39 30 6 180 greater than 0 and less than or equal to 20. 40 - 59 50 12 600 Thus, its mid-point is 10. The other intervals 60 - 79 70 25 1750 are interpreted accordingly. 80 - 99 90 5 450 Sum 50 3000 The arithmetic mean is x = 3000 / 50 = 60 marks. Note that if weights of size fi are suspended from a metre stick at the points xi, then the average is the centre of gravity of the distribution. Consequently, it is very sensitive to outlying values. x1 x2 f1 x xn fn f2 Equally the population should be homogenous for the average to be meaningful. For example, if we assume that the typical height of girls in a class is less than that of boys, then the average height of all students is neither representative of the girls or the boys. 3. The Mode Frequency 50 This is the value in the distribution that occurs most frequently. By common agreement, it is calculated from the histogram using linear interpolation on the modal class. 13 20 25 13 The various similar triangles in the diagram generate the common ratios. In our case, the mode is 60 + 13 / 33 (20) = 67.8 marks. 20 12 6 2 20 5 40 60 80 40 60 80 100 4. The Median Cumulative 50 This is the middle point of the distribution. It is used heavily in educational applications. If { x1, x2, … , xn } are the marks of students in a class, arranged in non-decreasing order, then 25.5 the median is the mark of the (n + 1)/2 student. It is often calculated from the ogive or cumulative frequency diagram. In our case, the median is 60 + 5.5 / 25 (20) = 64.4 marks. Frequency 20 100 Measures of Dispersion or Scattering Example 2. The following distribution has the same arithmetic mean as example 1, but the values are more dispersed. This illustrates the point that an average value on its own may not adequately describe a statistical distributions. To devise a formula that traps the degree to which a distribution is concentrated about the average, we consider the deviations of the values from the average. If the distribution is concentrated around the mean, then the deviations will be small, while if the distribution is very scattered, then the deviations will be large. The average of the squares of the deviations is called the variance and this is used as a measure of dispersion. The square root of the variance is called the standard deviation and has the same units of measurement as the original values and is the preferred measure of dispersion in many applications. Marks x Frequency f 10 30 50 70 90 Sums 6 8 6 15 15 50 fx 60 240 300 1050 1350 3000 x6 x5 x4 x3 x2 x1 x Variance & Standard Deviation s2 = VAR[X] = Average of the Squared Deviations = S f { Squared Deviations } / S f = S f { xi - x } 2 / S f = S f xi 2 / S f - x 2 , called the product moment formula. s = Standard Deviation = Variance Example 1 f x 2 10 6 30 12 50 25 70 5 90 50 fx 20 180 600 1750 450 3000 f x2 200 5400 30000 122500 40500 198600 VAR [X] = 198600 / 50 - (60) 2 = 372 marks2 Example 2 f x 6 10 8 30 6 50 15 70 15 90 50 fx 60 240 300 1050 1350 3000 f x2 600 7200 15000 73500 121500 217800 VAR [X] = 217800 / 50 - (60)2 = 756 marks2 Other Summary Statistics Skewness An important attribute of a statistical distribution relates to its degree of symmetry. The word “skew” means a tail, so that distributions that have a large tail of outlying values on the right-hand-side are called positively skewed or skewed to the right. The notion of negative skewness is defined similarly. A simple formula for skewness is Skewness = ( Mean - Mode ) / Standard Deviation which in the case of example 1 is: Skewness = (60 - 67.8) / 19.287 = - 0.4044. Coefficient of Variation This formula was devised to standardise the arithmetic mean so that comparisons can be drawn between different distributions.. However, it has not won universal acceptance. Coefficient of Variation = Mean / standard Deviation. Semi-Interquartile Range Just as the median corresponds to the 0.50 point in a distribution, the quartiles Q 1, Q2, Q3 correspond to the 0.25, 0.50 and 0.75 points. An alternative measure of dispersion is Semi-Interquartile Range = ( Q3 - Q1 ) / 2. Geometric Mean For data that is growing geometrically, such as economic data with a high inflation effect, an alternative to the the arithmetic mean is preferred. It involves getting the root to the power N = S f of a product of terms Geometric Mean = N x1f1 x2 f2 … xk fk