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Notes On Statistics

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Published in: Statistics
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Key Points on Statistics 

Arun M / Faridabad

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Teaches: Mathematics, IIT JEE Mains

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  1. March 17 2016 Statistics ( CLASS , CBSE BOARD ) KEY POINTS
  2. STATISTICS : Science of collection, presentation, analysis and interpretation of numerical data. PRIMARY DATA : Data collected by an investigator himself with a definite plan in his mind, is called Primary Data. This is also called Raw Data. SECONDARY DATA : Data obtained from published or unpublished sources are known as Secondary Data. RANGE : The difference between the maximum and minimum values of any set of observations is called its range. ARRAYED DATA ( or ARRAY ): Raw data when put in ascending or descending order is called an Array. VARIATE : Any character that can vary from one individual to another is called variable or variate. For example age, income, height etc. Variates are of two types - CONTINUOUS VARIATE : Quantities which can take all numerical values within a certain interval are called continuous variable. For example heights of different students, weights of a group of persons etc. DISCONTINUOUS or DISCRETE VARIATE : Variables which can take only a finite set of values are called discrete variables. For example number of students in a particular class, number of sections in a school etc. FREOUENCY : The number of times a variate occurs is called the frequency of the variate. Also the number of observation's corresponding to a particular class is called its frequency. CLASS : Each group into which the raw data is condensed is called a Class. For example O- 10, 10 - 20 etc. CLASS INTERVAL : The size of the Class is known as class interval. For example class interval for the class 0 10 is 10. CLASS LIMITS : Each class is bounded by two figures which are called the class limits. For example in class interval 10 20 these two figures are 10 and 20 respectively. Here 10 is the LOWER LIMIT and 20 is the UPPER LIMIT. INCLUSIVE DISTRIBUTION : In an inclusive distribution, the Upper limit of one class does not coincide with the lower limit of next class. For example classes 5 10, . represent an inclusive distribution. 11 16, 17 - 22, EXCLUSIVE DISTRIBUTION : In exclusive distribution, the Upper limit of one
  3. Class coincides with the lower limit of next class. For example classes 5 10 . represent an inclusive distribution. 10- 15, 15-20, TRUE CLASS LIMITS : In case of Exclusive classes the upper and lower limits are respectively called as its True Upper and True Lower Limits. CLASS SIZE : The difference between the True upper limit and True lower limit of a class is called the Size of the Class. CLASS MARK or MID VALUE : The value which lies midway between lower limits of a class is known as its mid value or class mark. Class Mark = ( Lower class limit + Upper class limit ) / 2 FINDING CLASS SIZE and CLASS BOUNDRIES FROM CLASS MARKS : Class Size = Difference of two consecutive class marks Half of the class size = Class size / 2 LOWER class Limit or Boundary = Class Mark — Half of Class size UPPER class Limit or Boundary = Class Mark + Half of Class size CHANGING INCLUSIVE DISTRIBUTION TO A EXCLUSIVE DISTRIBUTION : If a — b is a class interval in Inclusive method, then in Exclusive method it becomes ( a — h/2) - (b + h/2 ), where h = (lower limit of a class — upper limit of previous class ) / 2. CUMULATIVE FREOUENCY : In a discrete frequency distribution the cumulative frequency of a particular value of the variable is the total of all the frequencies of the values of the variable which are less than or equal to the particular value. In a grouped frequency distribution the cumulative frequency of a class is the total of all frequencies up to and including that particular class. Cumulative frequency are of two types viz. LESS THAN and MORE THAN ( or GREATER THAN ). For Less Than we add up the frequencies from the above and for More Than we add up the frequencies from below. Example - Given Table --- Marks No. of Students 0-10 3 10 - 20 12 20 - 30 36 30 - 40 76 40 - 50 97
  4. Less Than Cumulative Frequency Table --- Marks Obtained Less than 10 Less than 20 Less than 30 Less than 40 Less than 50 More Than Cumulative Frequency Table --- Marks Obtained More than 50 More than 40 More than 30 More than 20 More than 10 More than 0 Number of Students 3 15 51 127 224 Number of Students 97 173 209 221 224 GRAPHICAL REPRESENTATION OF DATA : This is another method of presenting the statistical data using squares, rectangles, circles etc. There are different types of graphs such as --- Bar Graphs Histogrms • Frequency Polygon ( Refer Text Book for details of these graphs ) ( While preparing the HISTOGRAMS if class intervals are UNEQUAL, than please adjust the Class frequencies using the following formula Adjusted frequency of a Class Interval = ( Minimum Class Size / Class Size ) x Frequency of the Class. ARITHMETIC MEAN ( or MEAN ) OF RAW AND UNGROUPED DATA : If x 1, x2, x3, are n observations, Then
  5. MEAN = (XI+ + x3+ +xn)/n — x/ n (E x is read as Sigma x ) ARITHMETIC MEAN OF DATA WITH FREOUENCY DISTRIBUTION: If Xl, X2, X3, . are n observations and fl, f2, f3,.. , fn are their corresponding Frequencies, Then MEAN = (flXl+ + f3X3+ +fnxn ) / ( fl+f2+f3+. (Where N = fl+f2+f3+. PROPERTIES OF ARITHMETIC MEAN : If each observation in a set of values is increased by any number a, then the Mean is also increased by a. If each observation in a set of values is decreased by any number a, then the Mean is also decreased by a. If each observation in a set of values is multiplied by any non-zero number a, then the Mean is also multiplied by a. If each observation in a set of values is divided by any non-zero number a, then the Mean is also divided by a. Algebraic sum of the deviations of a set of values from their Mean is zero. MEDIAN : The MIDDLE item of the arrayed data is called the Median. Calculation of Median of raw (arrayed) data is done as follows - If number of items ( n ) is ODD, then Median will be the value of [(n+l) /2] th. item. If number of items ( n ) is EVEN, then there will be two middle items, i.e. (n/2) th. item and [(n/2) + 1] th. item. Mean of these two items will be the Median. MODE : The value of the item which occurs maximum number of times is called Mode i.e. item which has maximum frequency is Mode. EMPIRICAL RELATION BETWEEN MODE MEDIAN and MODE : Mode = 3 Median — 2 Mean

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