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 fi 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 x1 x2 x xn from a metre stick at the points xi, then the average is the centre of gravity of the f1 fn distribution. Consequently, it is very sensitive f2 to outlying values. 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. :
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 fi 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 x1 x2 x xn from a metre stick at the points xi, then the average is the centre of gravity of the f1 fn distribution. Consequently, it is very sensitive f2 to outlying values. 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 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. The various similar triangles in the diagram generate the common ratios. In our case, the mode is 60 + 13 / 33 (20) = 67.8 marks. 4. The Median 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 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. :
3. The Mode 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. The various similar triangles in the diagram generate the common ratios. In our case, the mode is 60 + 13 / 33 (20) = 67.8 marks. 4. The Median 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 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. 50 Frequency 20 20 40 60 80 100 6 12 25 5 2 13 13 20 Cumulative
Frequency 100 80 60 40 20 50 25.5
Measures of Dispersion or Scattering Example 2. The following distribution has the same Marks Frequency arithmetic mean as example 1, but the values are more x f fx dispersed. This illustrates the point that an average value on its own may not adequately describe a 10 6 60 statistical distributions. 30 8 240 50 6 300 To devise a formula that traps the degree to which a 70 15 1050 distribution is concentrated about the average, we 90 15 1350 consider the deviations of the values from the average. Sums 50 3000 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. :
Measures of Dispersion or Scattering Example 2. The following distribution has the same Marks Frequency arithmetic mean as example 1, but the values are more x f fx dispersed. This illustrates the point that an average value on its own may not adequately describe a 10 6 60 statistical distributions. 30 8 240 50 6 300 To devise a formula that traps the degree to which a 70 15 1050 distribution is concentrated about the average, we 90 15 1350 consider the deviations of the values from the average. Sums 50 3000 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. x1 x2 x3 x4 x5 x6 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 = Ö Variance Example 1 Example 2 f x f x f x2 f x f x f x2 2 10 20 200 6 10 60 600 6 30 180 5400 8 30 240 7200 12 50 600 30000 6 50 300 15000 25 70 1750 122500 15 70 1050 73500 5 90 450 40500 15 90 1350 121500 50 3000 198600 50 3000 217800 VAR [X] = 198600 / 50 - (60) 2 VAR [X] = 217800 / 50 - (60)2 = 372 marks2 = 756 marks2 :
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 = Ö Variance Example 1 Example 2 f x f x f x2 f x f x f x2 2 10 20 200 6 10 60 600 6 30 180 5400 8 30 240 7200 12 50 600 30000 6 50 300 15000 25 70 1750 122500 15 70 1050 73500 5 90 450 40500 15 90 1350 121500 50 3000 198600 50 3000 217800 VAR [X] = 198600 / 50 - (60) 2 VAR [X] = 217800 / 50 - (60)2 = 372 marks2 = 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 Q1, 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 = NÖ x1f1 x2 f2 … xk fk :
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 Q1, 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 = NÖ x1f1 x2 f2 … xk fk