Basic numeracy-statistics
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Basic numeracy-statistics

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Basic numeracy, statistics

Basic numeracy, statistics

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Basic numeracy-statistics Basic numeracy-statistics Presentation Transcript

  • Click Here to Join Online Coaching Click Here www.upscportal.com Basic Numeracy
  • www.upscportal.com Click Here to Join Online Coaching Click Here Statistics Statistics The branch of Mathematics which deals with collection, classification and interpretation of data is called statistics. When used in the singular sense, statistics refers to the subject as a whole of science of statistical methods embodying the theory and techniques. When it is used in the plural sense, statistics refers to the data itself (ie, numerical facts collected in a systematic manner with some definite purpose in view, in any field of enquiry). The Frequency Table or the Frequency Distribution If the data is classified in a convenient way and presented in a table it is called frequency table or frequency distribution. Frequency: When the data is presented in a frequency table, the number of observations that fall in any particular class is called the frequency of that class.
  • www.upscportal.com Click Here to Join Online Coaching Click Here Class Limit: The starting and end values of each class are called “lower limit” and “upper limit” of that class respectively. Class-interval: The difference between the upper and lower boundary of a class is called the “class-interval” or “size of the class”. It can also be defined as the difference between the lower or upper limits or boundaries of two consecutive classes. Class Boundaries: The average of the upper limit of a class and the lower limit of the succeeding class is called the “upper boundary” of that class. The upper boundary of a class becomes the “lower boundary” of the next class. Range: The difference between the highest and the lowest observation of a data is called its range. Histogram: Pertaining to a frequency distribution, if the true limits of the classes are taken on the x-axis and the corresponding frequencies on the y-axis and adjacent rectangles are drawn, the diagram is called „histogram‟.
  • www.upscportal.com Click Here to Join Online Coaching Click Here Frequency Polygon and Frequency Curve: If the points pertaining to the mid values of the classes of a frequency distribution and the corresponding frequencies are plotted on a graph sheet and these points are joined by straight lines, the figure formed is called frequency polygon. If these points are joined by a smooth curve the figure formed is called frequency curve. Cumulative Frequency Curves: If the points pertaining to the boundaries of the classes of a frequency distribution and the corresponding cumulative frequencies are plotted on a graph sheet and they are joined by a smooth curve, the figure formed is called cumulative frequency curve. The figure formed with upper boundaries of the classes and the corresponding less than cumulative frequencies is called less than cumulative frequency curve. The figure formed with lower boundaries of the classes and the corresponding greater than cumulative frequencies is called greater than cumulative frequency curve.
  • www.upscportal.com Click Here to Join Online Coaching Click Here Arithmetic Mean (AM) or Mean 1. Arithmetic Mean of Ungrouped Data If x1, x2 , x3 ,... , xn are n values of a variable x, then arithmetic mean x is defined as Where = (x1 + x2 + x3 +...+ xn ) 2. Arithmetic Mean of Grouped Data Here, the mean may be computed by the following method Direct method If x1, x2 , x3 ,... , xn are n values of a variable x and fl, f2, f3,... fn are the corresponding frequencies, then Where, = f1x1 + f2x2 +...+ fnxn and N = f1 + f2 + f3 +...+ fn
  • www.upscportal.com Click Here to Join Online Coaching Click Here Median 1. Median of Ungrouped Data If x1, x2, ... , xn are n values of variable x arranged in order of increasing or decreasing magnitude then the middle-most value in this arrangement is called the median. If n is odd, then the median will be the ( n+1 / 2 ) value arranged in order of magnitude. In this case there will be one and only one value of the median. If n is even, then the data arranged in order of magnitude, will have 2 middle - most values ie, ( n / 2)th and ( n / 2 +1) values.
  • www.upscportal.com Click Here to Join Online Coaching Click Here 2. Median of Grouped Data If N is the number of observation, we first calculate N / 2 . Then from the cumulative frequency distribution, we determine the class in which observation lies. Let us name this as median class. We use the following formula for calculating the median. Median (M) = Where, l = lower boundary of the median class ie, the class where the ( N / 2 )th observation lies. N = total frequency F = cumulative frequency of a class preceding the median class. f = frequency of the median class. C = length of the class interval.
  • www.upscportal.com Click Here to Join Online Coaching Click Here Mode The mode or modal value of a distribution is that value of the variable for which the frequency is maximum. For a given data, mode may or may not exist. If mode exists for a given data, it may or may not be unique. Data having unique mode is called uni-modal. While the data having two modes is called bi-modal. 1. Mode of Ungrouped Data The observation with the highest frequency becomes the mode of the data. 2. Mode of Grouped Data Mode = Where, l = lower boundary of the modal class. f = frequency of the modal class. f1 = frequency of the class preceding the modal class. f2 = frequency of the class following the modal class. C = length of the class interval.
  • www.upscportal.com Click Here to Join Online Coaching Click Here Relation between Mean, Median and Mode 3(Median) – 2(Mean) = Mode Measure of Dispersion The dispersion of data is the measure of spreading (scatter) of the data about some central tendency. It is measured in the following types. (a) Range (b) Quartile Deviation (c) Mean Deviation (d) Standard Deviation (e) Variance. 1. Range Range is the difference of maximum and minimum values of the data. Range = Maximum value - Minimum value.
  • www.upscportal.com Click Here to Join Online Coaching Click Here 2. Quartile Deviation After arranging the data in the ascending order of magnitude we find QD = Where, for even observation Q1 = ( n/4)th observation and Q3 = ( 3n/4)th observation for odd observation. Q1 = (n+1/4)th observation and Q3 = 3(n+1)th / 4 observation. 3. Mean Deviation It is defined as the arithmetic mean of the absolute deviations of all the values taken about any central value. The mean deviation of a set of n numbers x1, x2, ... , xn is defined by MD =
  • www.upscportal.com Click Here to Join Online Coaching Click Here 4. Standard Deviation The standard deviation of a set of n numbers x1, x2 , ... , xn is defined by Standard deviation is denoted by the Greek letter (sigma). 5. Variance The variance of a set of it numbers x1, x2, ... , xn is defined as the square of the standard deviation.
  • www.upscportal.com Click Here to Join Online Coaching Click Here 6. Coefficient of Variation It is given by , where is the standard deviation and is the mean of the given observations. Example 1: If the numbers 3, 4, 6 and 8 occur with frequencies 1, 5, 2 and 4 respectively, then the arithmetic mean is Solution.
  • www.upscportal.com Click Here to Join Online Coaching Click Here Example 2: The weight of 20 students are given below: Weight (kg) 40 42 45 48 Number of students 6 8 4 2 Find the mean weight. Weight (kg) xi Frequency fi fixi 40 6 240 42 8 336 45 4 180 48 2 96 Total f = 20 fixi = 852 Solution
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