2.4 Bar charts and frequency polygons When the data are discrete and the frequencies refer to individual values, we display them graphically using a bar chart with heights of bars representing frequencies, or a frequency polygon in which only the tops of the bars are marked, and then these points are joined by straight lines. Bar charts are drawn with a gap between neighbouring bars so that they are easily distinguished from histograms. Frequency polygons are particularly useful for comparing two or more sets of data. Example Consider again the number of germinating seeds from Example 1. Using the frequency table constructed earlier, we can construct a Bar Chart and Frequency Polygon as follows. 4 6 0 2 Frequency 8 1 0 Barplot of the number of germinating seeds 85 86 87 88 89 90 91 92 93 94 Number of seeds 2 4 6 Frequency 8 1 0 Frequency polygon for the number of germinating seeds 86 88 90 92 94 Number of seeds n = 40 A third method which is sometimes used for qualitative data is called a pie chart. Here, a circle is divided into sectors whose areas, and hence angles are proportional to the frequencies in the 8 different categories. Pie charts should generally not be used for quantitative data – a bar chart or frequency polygon is almost always to be preferred. Whatever the form of the graph, it should be clearly labelled on each axis and a fully descriptive title should be given, together with the number of observations on which the graph is based. 2.5 Stem-and-leaf plots A good way to present both continuous and discrete data for sample sizes of less than 200 or so is to use a stem-and-leaf plot. This plot is similar to a bar chart or histogram, but contains more information. As with a histogram, we normally want 5–12 intervals of equal size which span the observations. However, for a stem-and-leaf plot, the widths of these intervals must be 0.2, 0.5 or 1.0 times a power of 10, and we are not free to choose the end-points of the bins. They are best explained in the context of an example. Example Recall again the seed germination Example 1. Since the data has a range of 9, an interval width of 2 (= 0:2 101) seems reasonable. To form the plot, draw a vertical line towards the left of the plotting area. On the left of this mark the interval boundaries in increasing order, noting only those digits that are common to all of the observations within the interval. This is called the stem of the plot. Next go through the observations one by one, noting down the next significant digit on the righthand side of the corresponding stem. 8 8 8 9 9 9 5 7 8 1 3 4 5 7 9 1 2 5 7 8 0 2 7 9 1 2 6 7 9 1 0 3 3 0 1 3 2 1 1 0 1 1 0 0 1 1 For example, the first stem contains any values of 84 and 85, the second stem contains any values of 86 and 87, and so on. The digits to the right of the vertical line are known as the leaves of the plot, and each digit is known as a leaf. Now re-draw the plot with all of the leaves corresponding to a particular stem ordered increasingly. At the top of the plot, mark the sample size, and at the bottom, mark the stem and leaf units. These are such that an observation corresponding to any leaf can be calculated as Observation = StemLabel StemUnits + LeafDigit LeafUnits to the nearest leaf unit. n = 40 8 8 8 9 9 9 5 6 8 0 2 4 5 7 8 0 2 5 7 9 0 2 7 9 0 2 7 7 9 0 0 3 3 1 3 1 3 1 1 1 Stem Units = 10 seeds, Leaf Units = 1 seed. 9 1 1 1 1 1 1 The main advantages of using a stem-and-leaf plot are that it shows the general shape of the data (like a bar chart or histogram), and that the all of the data can be recovered (to the nearest leaf unit). For example, we can see from the plot that there is only one value of 94, and three values of 89. Example A stem-and-leaf plot for the data from Example 5 is given below. An interval width of 0 :5 (= 0:5 100) is used. 1 2 2 3 3 4 5 0 6 0 5 1 n = 24 9 9 1 3 7 7 1 2 6 8 4 8 2 8 4 8 9 8 Stem Units = 1, Leaf Units = 0.1. Note that in this example, the data is given with more significant digits than can be displayed on the plot. Numbers should be “cut” rather than rounded to the nearest leaf unit in this case. For example, 1.97 is cut to 1.9, and then entered with a stem of 1 and a leaf of 9. It is not rounded to 2.0. 2.6 Summary Using the plots described in this section, we can gain an empirical understanding of the important features of the distribution of the data. Is the distribution symmetric or asymmetric about its central value? Are there any unusual or outlying observations, which are much larger or smaller than the main body of observations? Is the data multi-modal? That is, are there gaps or multiple peaks in the distribution of the data? Two peaks may imply that there are two different groups represented by the data. By putting plots side by side with the same scale, we may compare the distributions of different groups. MEASURES OF CENTRAL TENDENCY 3.1 Introduction The collected data as such are not suitable to draw conclusions about the mass from where it has been taken. Some inferences about the population can be drawn from the frequency distribution of the observed values. One of the important objectives of statistical analysis is to determine various numerical measures which describe the inherent characteristics of a frequency distribution. The averages are the measures which condense a huge unwieldy set of numerical data into single numerical value, are representative of the entire distribution. Hence, in finding a central value, the data are condensed into a single value around which most of the values tend to cluster. Commonly, such a value lies in the centre of the distribution and is termed as central tendency. 3.2 Measure of Central Tendency or Average One of the most important objectives of the statistical analysis is to get one single value that describes the characteristic of the entire mass of unwieldy data. Such a value is called the Central Value or an average. The word Average is very commonly used in everyday conversation. For example we talk of average milk yield of a cow, average fat content of milk , average height or life of an Indian, average income etc. When we say he is an average student what it means is that he is neither very good nor very bad, just a mediocre type of student. Similarly, when we talk of average size of butter or cheese packet being sold through a retail outlet what we mean is that the size of packet which is being sold to maximum number of individuals by the retail outlet that means it is the modal size. However, in statistics the term average has a different meaning. According to Croxton and Cowden An average value is a single value within the range of the data and is used to represent all of the values in the series. Since an average is within the range of the data, it is sometimes called a measure of central value. It may be defined as that value of a distribution which is considered as the most representative of the series or typical value for a group. Such a value is of great significance because It depicts the characteristic of the whole group It facilitates comparison Averages are sometime referred to as a measure of location since they enable us to locate the position or place of the distribution in question. Requisites of a good average It should be rigidly defined. It should be easy to understand and calculate. It should be based on all the observations. It should not be unduly affected by extreme observations. It should be suitable for further mathematical treatment. It should be least affected by fluctuations of sampling. The various measures of central tendency or averages are discussed below: 3.3 Arithmetic Mean Its value is obtained by adding together all the items and dividing it by the total number of observations. If X1, X2, . . . . . , Xn are n values of a variable X, then the arithmetic mean (A.M.) in case of raw data, is defined as In case of frequency distribution Xi X1 X2 X3 --- Xn fi f1 f2 f3 --- fn If the value X1 occurs f1 times, the value X2 occurs f2 times, the value Xn occurs fn times, then the arithmetic mean is given by where, 3.3.1 Arithmetic mean in case of grouped data In case of grouped data, arithmetic mean can be calculated by applying any of the following methods: i) Direct Method ii) Short cut Method iii) Step-deviation Method 3.3.1.1 Direct method Multiply each value of Xi (the mid value of the class) by the corresponding frequency fi. Obtain the sum of the products . Divide this sum of products by the total frequency (N) so as to get mean. Example 1: Find the mean from frequency distribution given in example 2 of Lesson 2. Solution: Prepare the following table and calculate arithmetic mean as follows: Class Interval Mid-value frequency (Xi) (fi) fi X i 1630-1730 1730-1830 1680 1780 17 19 28560 33820 1830-1930 1930-2030 2030-2130 2130-2230 1880 1980 2080 2180 23 16 14 7 43240 31680 29120 15260 2230-2330 2330-2430 2280 2380 2 2 4560 4760 Total 100 191000 3.3.1.2 Shortcut method (Change of Origin) If the values of X or/and f are large, the calculation of mean by direct method is quite tedious and time consuming. In such a case the calculations can be reduced to a great extent by using short cut method. This method consists in taking deviations of the given observations from any arbitrary value A. The formula for calculation of the arithmetic mean is Where A is arbitrary mean, di = Xi A i.e. deviation from the arbitrary or assumed mean. 3.3.1.3 Step-deviation method (Change of origin and scale) In case of grouped frequency distribution, with class intervals of equal magnitude, the calculations are further simplified by taking; where Xi is the mid value of the class and h is the common magnitude or width of the class intervals. So the formula for calculating mean is . The procedure is illustrated in the example 2. It will be seen that the answer in each of the three cases is the same. The step-deviation method is the most convenient method on account of simplified calculations. Example 2: Solve example 1 with short-cut and step-deviation method. Solution: Prepare the following table and calculate arithmetic mean as follows: a) Short-cut method: A=2080 b) Step-Deviation Method: A=2080, h=100 3.3.2 Mathematical properties of arithmetic mean i) The algebraic sum of the deviations of the given set of observations from their arithmetic mean is zero i.e. ii) If n1 and n2 are the sizes and pooled mean . are the respective means of two series then the of the combined series of size (n1+n2) observations is given by: iii) The sum of the squares of deviations of the given set of observations, when taken from their arithmetic mean, is minimum. 3.3.3 Merits of arithmetic mean The A.M. is rigidly defined It is based on all the observations It is easily calculated from the given data It is least affected by fluctuations of sampling It is suitable for further mathematical treatment. The average of two or more series can be obtained from the averages of the individual series. 3.3.4 Demerits of arithmetic mean The strongest drawback of arithmetic mean is that it is very much affected by extreme observations. In a distribution with open end classes the value of mean cannot be computed without making assumptions regarding the size of class. It can neither be located by inspection nor graphically. It cannot be used for qualitative type of data such as intelligence, honesty, flavor, overall acceptability of dairy product etc. Arithmetic mean cannot be obtained if a single observation is missing or lost. In a skewed distribution, usually arithmetic mean is not representative of the distribution. 3.4 Geometric Mean The geometric mean (usually denoted by G.M.) of a set of n observations is the nth root of their product. If X1, X2, . . . , Xn are n values of a variable X, none of them being zero, then the geometric mean, G.M. is defined by Thus logarithm of G.M. of a set of observations is the arithmetic mean of their logarithms. If X1, X2, . . . , Xn occurs f1, f2 . . . , fn times respectively then 3.4.1 Merits of geometric mean It is rigidly defined The G.M. is based on all observations of a series. It is not much affected by fluctuations of sampling. It is suitable for further mathematical treatment. Unlike arithmetic mean which has a bias for higher values, geometric mean has bias for smaller observations. As compared with Arithmetic mean, Geometric mean is affected to a lesser extent by extreme observations. 3.4.2 Demerits of geometric mean Computations are difficult It is not simple to understand It does not give equal weight to every item. It cannot be calculated if the number of negative values is odd as well as some value is zero. 3.4.3 Use of geometric mean It is most appropriate average when dealing with ratios, percentages and rate of increase between two periods. It is applied when increase or decrease in time is proportional e.g. growth of population is proportional to the time, increase in bacterial population is proportional to the time and rate of interest. Geometric Mean is used in the construction of Index numbers. 3.5 Harmonic Mean If X1, X2, . . . , Xn are n values of a variable X, then their Harmonic Mean, abbreviated as H.M. is defined by In other words Harmonic Mean is the reciprocal of the arithmetic mean of the reciprocals of the given observations. In case of grouped frequency distribution, harmonic mean is given by 3.5.1 Merits of harmonic mean It is rigidly defined The H.M. is based on all observations of a series. It is not much affected by fluctuations of sampling. It is suitable for further mathematical treatment. Since the reciprocals of the values of the variable are involved, it gives greater weight age to smaller observations and as such is not very much affected by one or two big observations. 3.5.2 Demerits of harmonic mean Computations are difficult and not simple to understand. It cannot be calculated if any one of the observations is zero. It is not a representative figure of the distribution unless the phenomenon requires greater weight age to be given to smaller values. 3.5.3 Use of harmonic mean H.M. is used in finding averages involving speed, time, price and ratios. It is useful for computing the average rate of increase of profits of a concern or average speed at which a journey has been performed or the average price at which an article has been sold. The rate usually indicates the relation between two different types of measuring units that can be expressed reciprocally. The H.M. is used for the problems about work, time and rate, where the amount of work is held constant and the average rate is required, or in problems about total cost, number of persons and per capita cost is called for or in problems of similar nature involving rates. The arithmetic mean (A.M.), the geometric mean (G.M.) and the harmonic mean (H.M.) of a series of n observations are connected by the relation A.M. ≥ G.M. ≥ H.M. The computation of G.M. and H.M. is illustrated in example 3. Example 3: Find G.M. and H.M. of data given in example 1. Solution: Prepare the following table and calculate G.M. and H.M. as follows: Class Interval Mid-value (Xi) 1630-1730 1730-1830 1830-1930 1930-2030 2030-2130 2130-2230 2230-2330 2330-2430 Total 1680 1780 1880 1980 2080 2180 2280 2380 frequency log Xi (fi) 17 19 23 16 14 7 2 2 100 3.2253 3.2504 3.2742 3.2967 3.3181 3.3385 3.3579 3.3766 fi log Xi 54.8303 61.7580 75.3056 52.7466 46.4529 23.3692 6.71587 6.75315 327.9316 0.0006 0.0006 0.0005 0.0005 0.0005 0.0005 0.0004 0.0004 0.0101 0.0107 0.0122 0.0081 0.0067 0.0032 0.0009 0.0008 0.0528 From example 2 and example 3 one can verify that A.M. ≥ G.M.≥ H.M. 3.6 Median (Positional Average) The median is defined the measure of the central value when arranged in ascending or descending order of magnitude. According to L.R. Connor The median is that value of the variate which divides the group in two equal parts, one part comprising all the values greater and the other, all values less than median. Thus, as against arithmetic mean which is based on all the items of the distribution, the median is only positional average i.e. its value depends on the position occupied by a value in the frequency distribution. 3.6.1 Calculation of median 3.6.1.1 Ungrouped data When the total numbers of observations are odd, then the median is the middle value after the observations are arranged in ascending or descending order of magnitude. If the number of observations is equal to n, then the value of ((n+1)/2)th item gives the value of median e.g. the median of 5 observations 65,69,52,58,45 i.e. 45,52,58,65,69 is 58. When the total number of observations is even then median is obtained as the arithmetic mean of the two middle observations after they are arranged in ascending or descending order of magnitude. If number of observations are say 2n, then the arithmetic average of nth and (n+1)st (central items) gives the value of median. If it is n then median is the arithmetic average of and � values e.g. the median of 6 observations 65,69,52,58,45,67 i.e. 45,52,58,65,67,69 is arithmetic mean of 58 and 65 which is equal to 61.5. 3.6.1.2 Grouped data Steps involved for its computation are: 1) Prepare less than cumulative frequency(c.f. ) distribution table 2) Find N/2. 3) Find cumulative frequency just greater than N/2 4) The class corresponding to step 3 contains the median value and is called the median class. The median for a grouped series is given by the following formula: Where: l is the lower limit of the median class f is the frequency of the median class h is width of the class interval c.f. is the cumulative frequency of the class preceding the median class. The computational procedure is illustrated in Example 3.4. 3.6.2 Merits of median It is rigidly defined. It is easily understood, very readily calculated and can exactly be located. It is readily obtained without the necessity of measuring all the objects. It is not affected by abnormally large or small values of the variable. Median can be computed while dealing with a distribution with open end classes. It can be determined by mere inspections and can be computed graphically. The median gives the best results in a study of direct qualitative measurements such as intelligence, honesty etc. 3.6.3 Demerits of median The median does not lend itself to algebraic treatment. The median of several series by combining the medians of the component series cannot be computed. Median being positional average is not based on each and every item of the observations. Median is relatively less stable than mean, particularly for small samples since it is affected more by fluctuations of sampling as compared to arithmetic mean. 3.7 Quartiles Quartiles are those values of the variate which divide the total frequency into four equal parts. Obviously there will be three such points Q1, Q2 and Q3 such that Q1≤ Q2 ≤ Q3 termed as the three quartiles. Q1is known as the lower or first quartile and is the value which has 25% of the items of the distribution below it and consequently 75 percent of the items are greater than it. Q3 is known as the upper or third quartile and has 75percent of the observations below it and consequently 25 percent of the observations above it. 3.8 Deciles Deciles are those values of the variate which divide the total frequency into 10 equal parts. The formula for obtaining jth Decile (Dj) in case of grouped frequency distribution is given as , j=1, 2, 3, ---, 9 3.9 Percentiles Percentiles are the values of the variates which divide the total frequency into 100 equal parts. The formula for obtaining kth Percentile (Pk) in case of grouped frequency distribution is given as , k=1, 2, 3, ---, 99. 3.9.1 Graphical method of locating position values The various partition values viz., median, quartiles, deciles and percentiles can be located graphically with the help of curve called the cumulative frequency curve or ogive. Draw a perpendicular from the point of the two ogives i.e. more than ogive and less than ogive on the x-axis, the foot of the perpendicular gives the value of median. The points corresponding to N/4, 3N/4, N/10,.., 9N/10, N/100,.., 99N/100 on y-axis with the foot values of the perpendicular on x-axis provide the value of Q1, Q3, D1, , D9, P1, ., P99. 3.10 Mode Mode is the value which occurs most in a set of observations and around which the other items of the set cluster densely. It is defined to be size of the variable which occurs most frequently or the point of maximum frequency or the point of greatest density. In other words mode is that value of observation for which the height of the ordinate is maximum. Modal value of the distribution is that value of the variate for which frequency is maximum. In the words of Croxton and Cowden The mode of a distribution is value at the point around which the items tend to be heavily concentrated. It may be regarded as the most typical value of a series of values. 3.10.1 Computation of mode In case of a frequency distribution, mode is the value of the variable corresponding to the maximum frequency. For a continuous frequency distribution, the class corresponding to maximum frequency is called the modal class. The mode is computed by the formula: Where l = lower limit of the modal class f1 is the frequency of the modal class. fo is the frequency of the class just preceding the modal class (pre-modal class). f2 is the frequency of the class just succeeding the modal class (post-modal class). h is the magnitude of the modal class. The computational procedure is illustrated in example 4. 3.10.2 Merits of mode It is easily understood. It is the most typical value and it is the most descriptive average. It is a positional average. It can be easily located by mere inspection of certain items. It can be easily determined from the graph. The extreme items have no effect provided they are not in the modal class. 3.10.3 Demerits of mode It is ill defined. A clearly defined mode does not always exist. The value of mode cannot always be determined. A distribution can be bimodal or multimodal. It is not based on all the observations of a series. It is not suitable of further mathematical treatment. As compared to mean, mode is affected to a greater extent by the fluctuations of sampling. Graphically mode can be located from the histogram of frequency distribution by making use of the rectangles erected on the modal, pre-modal and post modal classes. The method consists of following steps: a) Join the top right corner of the rectangle erected on the modal class with top left corner of the rectangle erected on the preceding class by means of a straight line. b) Join the top last corner of the rectangle erected on the modal class with top right corner of the rectangle erected on the succeeding class by means of a straight line. c) From the point of intersection of the lines in step (i) and (ii) above, draw a perpendicular to the X-axis. The abscissa of the point where this perpendicular meets the X-axis gives the modal value. Example 4: Find median, first quartile, third quartile and mode of the frequency distribution given in example 1 and obtain them graphically. Solution: Prepare the following table to calculate median, first quartile, third quartile and mode. Here N/2=50. Cumulative frequency greater than 50 is 59. Hence the median class is 18301930. N/4=25. Cumulative frequency greater than 25 is 36. Hence the first quartile class is 17301830. Q1 = 1730 + (25-17) = 1730 + 42.1053 = 1772.1053 3N/4=75. Cumulative frequency greater than 75 is 89. Hence the third quartile class is 20302130.