Foundation Notes
Foundation Notes
Arranging Data
In this Lesson we will get familiar with data and its various types. We will also discuss the methods of data collection. Then we will focus on various data presentation tools like table and graphs (like line chart, bar chart, pie diagram, pictogram and scatter diagram).
In this Lesson we will get familiar with frequency distribution and frequency polygon. We will also study the properties (skew ness and kurtosis) of frequency distribute on curve.
What is Data?
Data is a collection of related observations, facts or figures. A collection of data is called a data set, and each observation a data point. Example: Marks obtained by students in Introduction to Quantitative Methods course
Types of Data
Raw Data: Information before its systematic arrangement and analysis is called raw data. Useful inferences can be derived from the raw data by applying various statistical methods.Example: Sales data of a company for a year Data can be classified as:Published DataUnpublished Data Data that is already collected and published Data that is yet to be collected or printed RBI Bulletins, CMIE Reports Data collected by a shopkeeper regarding customer satisfaction and not published Primary Data Secondary Data First hand data collected by the way of sample survey or a census. Data collected from other available sources (collected by others)Observation, personal interview or questionnaires Company Annual Reports, Information from Internet Apart from this, data can also be classified along some characteristics of data like age, gender, education, income, etc. Some common methods of classification are Geographical, i.e. area-wise or region-wise Chronological, yearly data, quarterly data, monthly data, weekly data Qualitative, i.e., depending on characteristics By magnitude
Methods of Data Collection
Complete Enumeration (Census Survey or Census): - is a method in which the entire population is taken up and information is collected relating to all the units of the populationExample: Census conducted by Government of India every 10 years
This method gives accurate information but more resources (time, money and people) are required.
Sample Method: - is a method in which enumeration of a part of the population or universe is taken up and information is gathered regarding the selected part.Example: Checking only a few units from a production batch
The choice between the two methods of data collection depends on the factors like purpose of the enquiry, time available for making a decision, budget allocation, and the accuracy of data required for decision making.
Tables as Data Presentation Device
Tabular presentation is used to summarize or condense data. Tables help the managers to analyze the relationships and trends in the collected data.
Tabulation is the logical listing of related quantitative data in vertical columns and horizontal rows with sufficient explanatory and qualifying words, phrases and statements in the form of titles, headings and explanatory notes to make clear the full meaning, context and origin of the data.
Line Chart
In graphical presentation, the collected data is represented by various types of geometrical devices such as points, lines, bars, multi-dimensional figures, pictorials, etc. A graphical method is a non-quantitative form of presentation; the quantities are also indicated along with them. The magnitude of the data is depicted visually through the proportional size of the diagram or graph. Line chart is one of the effective graphical methods to depict the trend in a data. If the line is rising from left to right, then the data is showing an increasing trend and vice-versa.
Bar Chart
Bar charts use rectangles to present the data which is referred as bars. There are two types of bar charts vertical and horizontal. These diagrams are one-dimensional as the magnitude of the data is represented by length of the bar. The thickness or width of the bar has no relevance. The bars should be arranged from left to right. The given bar diagram shows the yearly sales of a company. Multiple bar diagram or compound bar diagrams are used to compare two or more sets of related data. This diagram is similar to the simple bar diagram, but bars in each set are placed together and gap is left between each set of bars. The given multiple bar diagram shows yearly export import values of a company.
Pie Diagram
Pie diagram is a circle divided into various segments and each segment represents the percentage contribution of various components to the total. Pie diagrams are used to compare many components simultaneously. For drawing a pie diagram it is necessary to express the value of each category as a percentage of the total. 3600 in a circle represent the whole (i.e., 100%) and 3.60 constitute 1% of the total. Degree of each part = Part 360/Total = Part 3.6 The pie diagram represents the share holding pattern of a company.
Pictogram
Pictograms represent the data in the form of pictures. The data is presented using appropriate pictures and their sizes indicate the magnitude of the data.
Scatter Diagram
Scatter diagram is used to study the correlation between two dependent variables. The scatter diagram is drawn by plotting the points on X and Y axis. When the points on the graph follow a pattern, it indicates high correlation and irregular pattern or behavior indicates low correlation.
Frequency Distribution
The table in which raw data is tabulated by dividing it into classes of convenient size and computing the number of data elements (or their fraction out of the total) falling within each pair of class boundary is called a frequency distribution table.Classes are groups of values having same characteristics of data. E.g. Employees of a company are grouped together on the basis of their ages. The range of values of a given class is called a class limits, and middle of a class interval is called class mark. For the class 25-29, 25 and 29 are called as class limits, 27 is the class mark and 30-25 = 5 is the class interval. A cumulative frequency distribution is a tabular display of data showing how many observations lie above, or below, certain values.
Construction of Frequency Distribution
To construct a frequency distribution, the data is to be divided into groups of similar intervals. Then the number of data points that fall into each group has to be recorded against each group. Frequency distributions can be constructed with classes of qualitative attributes. The classification can be either quantitative or qualitative and either discrete or continuous classes.
Histogram
A histogram is a series of rectangles, the width of each being proportional to the range of values within a class and height being proportional to the number of items falling in the class. The widths of the bars are uniform when the widths of classes in a frequency distribution are equal.When a histogram is constructed using relative frequency, it is called a relative frequency histogram. While the absolute histogram represents the number of data items, the relative frequency histogram shows the relative size of each class with the total.
Frequency Polygon
For constructing a frequency polygon, the frequencies are marked on the vertical axis and the values of variables (that are being studied) are taken on the horizontal axis. Dots are put on the graph against the class marks to represent the frequencies. These dots are connected by drawing straight lines, this forms a frequency polygon. When the straight line are smoothed by adding classes and data points, is called a frequency curve.
Frequency polygons represent graphically both simple and relative frequency distributions.
Ogive
Frequency Distribution Table
The Less than Ogive Curve for the above Frequency Distribution is:
When the cumulative frequencies are plotted on a graph we get an Ogive. Ogive are of two types less than ogive and more than ogive. The more than ogive slopes down and to the right whereas the less than ogive slopes up and to the right.
Skew ness
Skew ness and Kurtosis are the two characteristics of data sets that provide useful trends and patterns in the data represented as frequency distribution curves. Skew ness is the extent to which a distribution of data points is concentrated at one end or the other; or the lack of symmetry in the curve. The curves representing the data points in the data set can be of two types: Symmetrical curves :- A curve is said to be symmetrical when a vertical line drawn from the center of the curve to the X-axis divides the area under the curve into equal parts. Skewed curves (positively or negatively skewed):-A curve is said to be skewed when the values in the frequency distribution are concentrated more towards the left or right side of the curve i.e. the values are not equally distributed from the center of the curve. A curve is said to be positively skewed when the tail of the curve is more stretched towards the right side. It is said to be negatively skewed when the tail is more stretched towards the left side.
Kurtosis
Kurtosis is the degree of peak ness of a distribution of points i.e. Kurtosis measures the peaked ness of a distribution. Two curves with same central location and dispersion may have different degrees of kurtosis.
Summary
Data is a collection of related observations, facts or figures.
Data can be categorized into published data and unpublished data.
Data collection is done in two ways complete enumeration and sample method.
Data is systematically and clearly represented in the form of tables and graphs.
Line charts, bar charts, pie diagram, scatter diagram are some of the tools that are used to graphically represent the data.
A frequency distribution is a tabular form that organizes data into classes.
Frequency polygons are graphical representation of frequency tables.
Skewness is the lack of symmetry in a curve
Kurtosis is degree of peaked ness of a distribution of points.
Measure of central Tendency
In this Lesson we will get familiar with measures of central tendency. We will study the objectives of averaging and requisites of good average. We will also focus on other types of averages like arithmetic mean, weighted arithmetic mean, geometric mean, harmonic mean, median and mode.
Objectives of Averaging
To find out one value that represents the whole mass of data If the researcher knows the average value of the data, then he need not study each and every data point in the data set. To enable comparison Averages act as a common denominator for comparing two or more sets of data. To establish relationship Averages play a major role in establishing relationships between separate groups in quantitative terms. To derive inferences about a universe from a sample The average calculated from a sample data give a reliable idea about the average of the entire universe. To aid decision-making Averages act as benchmarks or standards for managerial control and decision-making.
Requisites of Good Average
An ideal average should have the following characteristics: Should be rigidly defined Should be mathematically expressed (Have a mathematical formula) Should be readily comprehensible and easy to calculate Should be calculated based on all the observations Should be least affected by extreme fluctuations in sampling data. Should be suitable for further mathematical treatment. In addition to the above requisites, a good average should also retain maximum characteristics of the data, it should be a nearest value to all the data elements. Averages should be calculated for homogeneous data i.e. ages, sales etc.
Types of Averages
Averages are basically divided into two types: Mathematical averages and positional averages. The mathematical averages are arithmetic mean, geometric mean and harmonic mean. The positional averages are median and mode.
Arithmetic Mean
The mean of a sample containing n observations is given by = (1/n) (x1 +x2 +...+ xn) =(x/n=(1/n)
where, is sample meann is the number of elements When the mean is calculated for the entire population it is known as population arithmetic mean (). N is the number of elements (observations) in the population. Then=( x/NExample: The height of five friends is A=5.6, B=5.9, C=5.8, D=6.0, E=5.7. What is their average height?=(x / n=(5.6 + 5.9 + 5.8 + 6.0 + 5.7) / 5= 5.8 Grouped Data Calculate the mid-point of each class Mid-point = (Lower Limit + Upper Limit) / 2 Multiply each mid-point by frequency of observations in the corresponding class (f.x)
= ( (f x )/n f = Number of observations in each class x = class mark (mid point of each class) n = Number of observations in the sample
Class Frequency 21-25 38 26-30 30 31-35 35 36-40 25 41-45 15 46-50 12 51-55 3 56-60 2
Class Frequency (f) Class Mark (x) f x 21-25 38 23 874 26-30 30 28 840 36-40 25 38 950 41-45 15 43 645 46-50 12 48 576 51-55 3 53 159 56-60 2 58 116 n = 160 f x = 5315 =( (f x )/n=
=33.218Short-cut Method Locate an assumed mean. Assign a code value zero to the class containing assumed mean Assign negative integers as codes to the classes with values smaller than assumed mean and positive integers to the classes with values larger than assumed mean = x0 + w ( (u f)/n Where,
= Mean X0 = value of the class mark assigned the code 0 w = numerical width of the class interval U = code assigned to each class F = frequency of the class (number of observations) N= total number of observations in the sample Example: We will solve the previous example by the short-cut method.Class Class Mark (X) Code (u) Frequency (f) u f 21-25 23 -3 38 -114 26-30 28 -2 30 -60 31-35 33 -1 35 -35 36-40 38 0 25 0 41-45 43 1 15 15 46-50 48 2 12 24 51-55 53 3 3 9 56-60 58 4 2 8 -153 =x0 + w ( (u f)/n= 38 + 5 -153 / 160 = 33.218
Weighted Arithmetic Mean
The weighted mean is calculated taking into account the relative importance of each of the values to the total value. The formula for calculating the weighted average is:w =((w x)/ Sw Where,w =symbol for weighted meanW=weight allocated to each observation(wx)=sum of each weight multiplied by that elementSw =sum of all the weightsExample: Class of Labour Wage per hour (x) (Rs) Labour hours per unit Product 1 Product 2 Unskilled Semiskilled Skilled 10 15 20 2 3 5 6 2 1 The labor cost / hour for Product 1 is given byxw =( (wx) / Sw=
= Rs 16.5/1=Rs. 16.5 per hourSimilarly for labor cost / hour for Product 2 is given byxw= ( (wx) / Sw
=
= = Rs. 12.22 per hour
Median
The median is the middle value of a series arranged in ascending or descending order. The median is the 50th percentile value below which 50% of the values in the sample fall. Ungrouped Data If the dataset contains an odd number of items, the middle item of the dataset is the median If the dataset contains an even number of items, the average of the two middle items is the median If the total of the frequencies is odd, say n, then value of (n+1)/2th item gives the median If the total of the frequencies is even, say, 2n, then the arithmetic mean of nth and(n + 1)th gives the median Example: A fruit vendor recorded the sales of oranges for a week. Day Sunday Monday Tuesday Wednesday Thursday Friday Saturday Number of oranges 280 240 250 220 270 225 265 What is the median number of oranges sold in that week? Solution: First arrange the data in ascending order DaysNumber of orangesWednesday220Friday225Monday240Tuesday250Saturday265Thursday270Sunday280
The dataset contains 7 data points, so the median is given by the middle item, i.e. item number 4. Thus the median for the given data is 250. Grouped Data To find the median for grouped data, first we need to identify the median class. It is assumed that the items are evenly spaced over the entire class interval. Then by interpolation median is calculated as Median =
W + Lm where,
Lm = lower limit of the median class fm = frequency of the median class F = cumulative frequency up to the lower limit of the median class W=width of the class intervalN=total frequencyExample:Class Frequency Cumulative Frequency 101-200 6 6 201-300 12 18 301-400 18 36 401-500 27 63 501-600 21 84 601-700 17 101 701-800 15 116 801-900 11 127 901-1000 9 136 The total frequency of the data N = 136, thus median is given by item. i.e. 68.5th item,which lies in 501-600 class. The median class is 501-600 class.Lm=501,N=136,F=63,fm=21,W=100Median =
W + Lm
=
= (0.21428 100) + 501 = 522.428
Mode
Mode is defined as the value of the variable which occurs most frequently in the data set. When the data is grouped in a frequency distribution the manager must assume that the mode is located in the class with highest frequency. The mode can be found using the following equation.
Mode, Mo = Lmo +
w Where, Lmo = lower limit of the modal class d1 = frequency of the modal class - the frequency of the class just below it d2 = frequency of the modal class - the frequency of the class just above it w = width of the modal class
Advantages and Disadvantages
In case of a symmetrical distribution, mean, median and mode coincide. In case of a moderately asymmetrical, the mean, median and mode are related in the following mannerMode = 3 Median - 2 Mean
Summary
We analyze the data statistically to calculate the average point of the data.
The average point of the data that is located centrally is called as the measure of central tendency.
There are two types of averages mathematical averages Arithmetic mean, Geometric mean and Harmonic mean and Positional averages Median and mode.
Measure Of Dispersion
In this Lesson we will get familiar with what is dispersion. We will study a few measures of dispersion namely range, quartile deviation and mean deviation along with their merits and limitations. In this session we will discuss the calculation of these measures for ungrouped and grouped data. To study measures of dispersion: variance and standard deviation To study Bienayme Chebyshevs rule
Dispersion
Dispersion of a dataset measures the variability of the data or how data is distributed in a dataset. When the dispersion is measured in terms of the difference between two values selected from the data set, it is called as distance measure. E.g. The range, the interquartile range and quartile deviation When the dispersion is measured in terms of the average deviation from some measure of central tendency, it is called as average deviation measure. E.g. Mean Deviation, Variance and Standard Deviation
The Range
For ungrouped data, range is defined as the difference between the value of the smallest observation and the value of the largest observation present in the distribution. Range = Largest Value Smallest Value For grouped data, range is defined as the difference between the upper limit of the highest class and the lower limit of the smallest class. Range = Upper limit of the highest class - Lower limit of the lowest class Coefficient of range is relative measure of range and is used for comparing observations in different units. For example, a physical trainer cannot compare the range of the weights of employees with range of their heights as the range of weights would be in kilograms and that of heights in centimeters. Coefficient of Range=
=
Example: Calculate range and coefficient of range for the given data:45, 67, 87, 55, 74, 81Range = Largest Value Smallest Value
= 87 45 = 42Coefficient of Range =
=
= 0.318 Example: Calculate range and coefficient of range for the given data:Class 0-10 11-20 21-30 31-40 41-50 Frequency 5 7 10 8 9
Range = Upper limit of the highest class - Lower limit of the lowest class = 50 0 = 50
Coefficient of Range =
=
= 1 Merits:1Range is simple to understand and easy to calculate.2Range is the quickest way to get a measure of dispersion, although it is not accurate. Limitations:1.It is not based on all the observations in the data. It is computed based on the highest and the lowest values and ignores the nature of dispersion among other values of observations in the data set. 2.It is influenced by extreme values and hence fluctuates from sample to sample of a population, even though the values that fall in between the highest and lowest values are similar. 3.Range cannot be computed for frequency distributions with open-end classes.4.Range fails to explain about the character of the distribution within two extreme observations (i.e. L and S)5.Range is unreliable as a measure of dispersion of the values within a distribution.Uses: The quality control experts analyze the dispersion of a products quality. If the dispersion is more, that means the quality keeps changing, if the dispersion is less then the quality remains more or less the same.
Financial analysts are concerned about the dispersion of a firms earnings. Widely dispersed earnings, those varying from extremely high to low, indicate a higher risk to stockholders and creditors than do earnings remaining relatively stable.
Quartile Deviation
Interquartile RangeThe range calculated on the basis of middle 50% of the observations is called as interquartile range. This interquartile range is calculated from observations obtained after discarding one quartile of the observations at the lower end and another quartile of the observations at the upper end of the distribution. Thus, interquartile range is the difference between the third quartile and the first quartile. Interquartile range=Q3-Q1 Quartile DeviationQuartile deviation is defined as one half of the interquartile range. Quartile deviation gives the average value by which the two quartiles differ from the median. In symmetrical distribution, the quartiles Q3 and Q1 are equidistant from the median i.e. Median - Q1 = Q3 Median Quartile deviation (Q.D.)=
The relative measure of quartile deviation is called coefficient of quartile deviation. It can be used to compare the degree of variation in different distributions. Coefficient of Q.D=
For Ungrouped Data Lower quartile (Q1) = observation
Upper quartile (Q3)= observationWhere,N=total number of observationsExample: The sales figures of a company are given below. Calculate the quartile deviation for the sales data.
Month & Year April 02 May 02 June 02 July 02 Aug. 02 Sept. 02 Oct.02 Sales (in Rs. 000) 15.6 16.3 18.1 19.5 20.4 21.5 22.7 Q1 = =
= 2 Q3 =
=
= 6 The 2nd observation is 16.3 and the 6th observation is 21.5
Quartile deviation (Q.D.)=
=
= 2.6 For Grouped Data Q1 =
Q3 =
Where,
L1 = the lower boundary of the first quartile class (Q1) L3 = the lower boundary of the third quartile class (Q3) N = Total cumulative frequency f = Frequency of the quartile class h = Class interval (width) C = Cumulative frequency of the class just above the quartile class
Example: The wages of employees are given below. Calculate the quartile deviation and coefficient of quartile deviation.Wages 1501-2500 2501-3500 3501-4500 4501-5500 5501-6500 No. of Employees 3 10 15 12 2 Wages No. of Employees Cumulative Frequency 1501-2500 3 3 2501-350010 13 3501-4500 15 28 4501-5500 12 40 5501-6500 2 42 Cumulative Frequency Table Q1 =
=
= 10.75th observation This observation will fall in class (2501-3500)
L1 = 2501, C = 3, f = 10, h = 1000
Q1 =
=
= 3251
Q3 =
=
= 32.25th observation This observation will fall in class (4501-5500)
L3=4501,C=28,f=12Q3=
=
= 4792.667 Quartile Deviation =
=
= 770.833
Coefficient of Q.D. =
=
= 0.787 Merits: Q.D can be used as a measure of variation for open-ended distributions. Q.D. is a better measure of variation for highly skewed distribution or distribution with extreme values as Q.D. is not affected by the presence of extreme values. Limitations: As the Q.D is calculated using only 50% of the total observations, it cannot be regarded as a good measure of variation. Q.D. is not a real measure of variation as it does not measure the scatter of observations from the average. Q.D. is only a positional average.
Mean Deviation
Calculation of mean deviation for ungrouped data
Calculate the sample mean Subtract the mean from every value in the data set and ignore the positive or negative signs Add all the differences and divide the sum by the number of items in the sample Absolute Mean Deviation =
(for a sample) Example: The maximum day temperature was recorded for 10 days. Calculate the absolute mean deviation.
Day 1 2 3 4 5 6 7 8 9 10 Temperature (oC) 25.0 24.8 25.2 24.6 24.0 23.7 23.3 23.0 22.7 22.5
Day Temperature (oC) Deviation from mean (x - )Absolute deviation 1 25.0 1.12 1.12 2 24.8 0.92 0.92 3 25.2 1.32 1.32 4 24.6 0.72 0.72 5 24.0 0.12 0.12 6 23.7 -0.18 0.18 7 23.3 -0.58 0.58 8 23 -0.88 0.88 9 22.7 -1.18 1.18 10 22.5 -1.38 1.38 N=10 (x= 238.8 = 8.4 Mean( )=
=
=23.88
Absolute Mean deviation =
=
= 0.84 Example: Calculate mean deviation for the given data.
Class 0-200 201-400 401-600 601-800 801-1000 Frequency 32 108 67 28 14 Solution:
Class Interval Frequency (f) Mid-value of class interval (X) 0-200 32 100 3200 307.0879 9826.8128 201-400 108 300.5 32454 106.5879 11511.493 401-600 67 500.1 33506.7 93.0121 6231.8107 601-800 28 700.1 19602.8 293.0121 8204.3388 801-1000 14 900.1 12601.4 493.0121 6902.1694 N= =249 = 2500.8 = 101364.9 =42676.624 Hint: Use MS Excel to demonstrate the example
=
= 407.0879 Absolute Mean Deviation =
=
= 171.3920 Merits: Absolute mean deviation is simple and easy to understand. Absolute mean deviation is a more comprehensive measure of dispersion as it is dependent on all observations of a distribution. As it is obtained by taking the average of the deviations of every observation from the mean, it is a true measure of dispersion. Limitations: Absolute Mean deviation is less reliable as it is the arithmetic mean of the absolute values (ignoring the positive and negative signs). Absolute Mean deviation is not conducive to further algebraic treatment. Absolute Mean deviation cannot be computed for distributions with open-end classes.
Variance (2)
Steps for calculating variance for ungrouped data: Calculate the sample mean
Subtract the mean from every value in the data set and square the difference Add all the differences and divide the sum by the total number of items in the sample =
Steps for calculating variance for grouped data: Calculate the Sample mean
= ( (f x )/fWhere x is the mid-point of the class and f is the frequency of the class
Calculate the difference between the sample mean and the mid-point of the class and square the difference Multiply the frequency of the class and the squared difference. Add all the products and divide the sum by the total frequency
=
Standard Deviation ()
Standard deviation is the square root of the average of the squared distances of the observations from the mean (i.e. square root the variance). Standard deviation for ungrouped data,=Standard deviation for grouped data,
Properties of Standard Deviation
Standard Deviation is independent of change of origin The value of standard deviation remains the same, if in a series each of the observation is increased or decreased by a constant quantity. For example, for the observations 3, 10 and 12 = 8.33,
= 3.85If we increase the value of each observation by 4.5 we get the observations 7.5, 14.5 and 16.5.
Now=12.833 and
=
= 3.859 Hence although has increased by 4.5, remains the same. Standard Deviation is dependent on the change of scale For a given series, if each observation is multiplied or divided by a constant quantity standard deviation will also be similarly affected. Suppose we multiply each observation by 6, the observations become 18, 60 and 72. = 50 =
= 23.152 which is nothing but the earlier , 3.859 6. Standard deviation is the minimum root-mean- square deviation The sum of the squares of the deviations of items of any series from a value other than the arithmetic mean would always be greater. We know that it is possible to compute combined mean of two or more groups, it is also possible to compute combined standard deviation of two or more groups. Combined standard deviation denoted by is computed as follows:
Where,
=standard deviation of first group
=standard deviation of second groupd1 =1 - d2 =2 - =(n11 + n22 ) / n1 + n2
Coefficient of Variation
The coefficient of variation is a measure of relative dispersion and is given by Coefficient of variation (%) =
100 The coefficient of variation measures the spread of a set of data as a proportion of its mean. It is used in problem situations where we want to compare the variability, homogeneity, stability, uniformity and consistency of two or more data sets. The data set for which the coefficient of variation is greater is said to be more variable i.e. less consistent or less homogeneous. On the other hand, if the coefficient of variation is less it is said to be less variable i.e., more consistent or more homogeneous. Example 1: Find the standard deviation and the coefficient of variance for the given data. 1 2 3 4 5 6 xi 15 13 17 16 18 20 xi (xi- ) (xi- )2 15 -1.5 2.25 13 -3.5 12.25 17 0.5 0.25 16 -0.5 0.25 18 1.5 2.25 20 3.5 12.25 Sum = 99 29.50 =
= 16.5 = 4.91 =
= 2.21 Coefficient of variation (%) =
100 =
= 13.429 Example 2: Find the standard deviation and the coefficient of variance for the given data.Class 0-10 11-20 21-30 31-40 41-50 51-60 Frequency 6 8 13 15 18 20
Class Frequency (f) Mid-point (x) f x x- (x- )2 f(x- )2 0-10 6 5 30 -31.8375 1013.6264 6081.7584 11-20 8 15.5 124 -21.3375 455.2889 3642.3112 21-30 13 25.5 331.5 -11.3375 128.5389 1671.0057 31-40 15 35.5 532.5 -1.3375 1.7889 26.8335 41-50 18 45.5 819 8.6625 75.0389 1350.7002 51-60 20 55.5 1110 18.6625 348.2889 6965.778 Sum 80 2947 -38.525 2022.5709 19738.387
=
= 246.7298 =
=15.7076 Coefficient of variation (%) =
100 =
= 42.6402
