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Statistics

Statistics. 9th Grade – Maths. INTRODUCTION. What is Statistics ? Statistics  is the study of the collection, organization, analysis, interpretation, and presentation of data . What is the need to learn Statistics ?

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Statistics

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  1. Statistics 9th Grade – Maths 9th - Maths

  2. INTRODUCTION What is Statistics? Statistics  is the study of the collection, organization, analysis, interpretation, and presentation of data. Whatis the need to learnStatistics? • Statistics helps in providing a better understanding and exact description of a phenomenon of nature. • Statistical helps in proper and efficient planning of a statistical inquiry in any field of study. • Statistical helps in collecting an appropriate quantitative data. • Statistics helps in presenting complex data in a suitable tabular, diagrammatic and graphic form for an easy and clear comprehension of the data. • Statistics helps in understanding the nature and pattern of variabilityof a phenomenon through quantitative observations. • Statistics helps in drawing valid inference, along with a measure of their reliability about the population parameters from the sample data. 9th - Maths

  3. Frequency Distribution Table • A tabular arrangement of data (scores) by classes is called a Frequency Distribution Table or Frequency Table (F.D.T.) • Exclusive type: Here the upper limit of one class OVERLAPS with the lower limit of the next class. Example: • Inclusive type: Here the upper limit of one class DOES NOT OVERLAP with the lower limit of the next class. Example: 9th - Maths

  4. Mean • To recap: For a data set, the arithmetic MEAN is equal to the sum of the values divided by the number of values. • i.e. • How to calculate mean for a given FDT, when actual scores are not known? • Consider the table: 9th - Maths

  5. Steps to calculate Mean • Step 1: 1st class interval is 1-5, and frequency is 2. The representative score (x) of the class is the mid-score 1, 2, 3, 4, 5 Another way of calculating representative score: 9th - Maths

  6. Steps to calculate Mean • Step 2: The sum of n scores in C.I is Where is the frequency and is the representative score. For C.I. 1-5, the sum is be For C.I. 6-10, the sum is be and so on. Now the table looks like this: 9th - Maths

  7. Steps to calculate Mean • Step 3: Now you know the sum of all scores , and also the number of scores . Now 9th - Maths

  8. Median • MEDIAN is described as the numerical value separating the higher half of a sample from the lower half. • Simply put, Median is the middle most score when the given scores are arranged in an order. • When the number of scores is odd, there is only 1 middle number, which is the median. • When the number of scores is even, there are 2 middle numbers. In that case, Median is calculated as the average of the 2 middlemost numbers 9th - Maths

  9. Median Examples. • Find the median of the following scores: 20, 24, 43, 30, 40, 28, 32 Step 1: Arranging the above scores in increasing order: 20, 24, 28, 30, 32, 40, 43 Step 2: Find the middle most score: Median = 30 Here N is 7, an odd number. So there is only 1 middlemost number 20, 24, 28, 30, 32, 40, 43 9th - Maths

  10. Median • How to find a median when grouped data is given? • As the actual scores are not given, how to find the middlemost number? • Consider the following FDT: 9th - Maths

  11. Calculate Median of FDT Step 1: N = 20. Therefore the midscore must be . If 20 scores are arranged in an order, then the midscore must be average of 10th and 11th score. Step 2: Add cumulative frequency to the FDT. 9th - Maths

  12. Calculate Median of FDT Step 1: N = 20. Therefore the midscore must be . If 20 scores are arranged in an order, then the midscore must be average of 10th and 11th score. Step 2: • Add cumulative frequency to the FDT. • The 10th and 11th score lies in C.I. 11-15. • 11-15 is called the Median Class. • The lower limit of Median interval is 10.5 • Size if C.I. is 5 9th - Maths

  13. Calculate Median of FDT Step 3: • We need 4 more scores (10 – 6) out of 6 scores from the Median Class. • of the Class interval 11 – 15 • This needs to be added to lower limit. 9th - Maths

  14. Mode • The mode is the value that appears most often in a set of data. • Find the mode for the following scores: 20, 22, 30, 28, 22, 36, 28, 22, 30 We know the most repeated score is 22. 9th - Maths

  15. Calculating Mode for an FDT Find the mode of the FDT given. Crude Mode • We see that C.I. 11-15 has the highest frequency, 6. • The midpoint of this C.I. is 13. • Mode is the midpoint of the C.I. having the maximum frequency. 9th - Maths

  16. Calculating Mode for an FDT True Mode We have already calculate median and mean of the same FDT. Mean=13.75=13.8(round to first decimal) 9th - Maths

  17. Dispersion Various methods to find the Dispersion • Range • Quartile Deviation • Mean Deviation • Standard Deviation 9th - Maths

  18. Dispersion • Dispersion gives us an idea of the homogeneity or heterogeneity of the distribution. Measures of Dispersion • Range • Quartile Deviation • Mean Deviation • Standard Deviation We will see the first three methods in details 9th - Maths

  19. Dispersion - Range • Range is the difference between two extreme scores of the distribution. • If ‘H’ is the highest score, and ‘L’ is the lowest score, then Range = H - L 9th - Maths

  20. Dispersion - Range 9th - Maths

  21. Dispersion - Range • In the table, the average wage for all 3 factories is 2600. • Calculate Range for all three factories (Range = H – L) • Factory A: H = 2600, L = 2600 • Factory B: H = 2960, L = 2000 • Factory C: H = 3500, L = 1500 9th - Maths

  22. Dispersion - Range Conclusion • There is no variation in wage of workers of factory A. • There is little variation in wages of Factory B. • The wages in Factory C varies widely. This shows that mean income is NOT SUFFICIENT to know how income is distributed. 9th - Maths

  23. Dispersion - Range • Coefficient of Range: The ratio of difference between highest and lowest score to the sum of highest and lowest score. • Calculating Coefficients for the three factories: • Factory A: Coefficient of Range = • Factory B: Coefficient of Range = • Factory A: Coefficient of Range = 9th - Maths

  24. Dispersion - Range Conclusion based on Coefficient of Range • Coefficient is always between 0 and 1. • As the value approaches 0, the distribution of scores becomes more and more consistent. • It helps to compare the performance of two or more groups. 9th - Maths

  25. Dispersion - Range Merits • Range is simplest to understand and easiest to compute. • Takes minimum time to calculate the range. Limitation • Not based on each and every observation. • Range cannot tell us about the character of distribution. Two distribution having same range can have different characteristics. Uses • Useful in studying the variations in stock/share prices, or other commodities like Gold. • Weather Forecast. 9th - Maths

  26. Dispersion – Quartile Deviation Lowest Score Quartile Deviation is also called Semi-InterQuartile Range, ‘Q’. Quartile Deviation is based on the central 50% of the scores. What is a Quartile? Divide the distribution of scores in 4 equal parts. There will be 3 such scores (Q1, Q2,Q3). These scores are called Quartiles. 1st Quartile Q1 25% of score 2ndQuartile Q2 Interquartile Range 25% of score 3rdQuartile Q3 N = number of scores Highest Score 9th - Maths

  27. Dispersion – Quartile Deviation N = 15 = = 36 = = 68 9th - Maths

  28. Dispersion – Quartile Deviation Step 1: To find Q1 • From C.F., 12.5th lies in C.I. 19-21. • Lower Limits of CI 19-21 = 18.5 (L) • C.F below to C.l. 19-21 (F) = 10 • Frequency of CI 19-21 (f) = 5 • Size of the class interval (i) = 3 9th - Maths

  29. Dispersion – Quartile Deviation Step 2: To find Q1 • Substituting the values: Step 3: To find Q3 • 37.5th score lies in C.I 31-33 • Lower limit of C.I. = 30.5 (L) • Frequency of C.I. 31-33 (f) = 5 Step 4: To find Q3 • Substituting the values: 9th - Maths

  30. Dispersion – Quartile Deviation Step 2: To find Q1 9th - Maths

  31. Dispersion – Quartile Deviation Merits • Merits of Quartile Deviation • It is easy Lo understand and compute. • It ís not affected by the extreme scores. • It is a useful measure when it is desired to know the variability in the central half of the data Demerits • It ignores 50% of the scores i.e., first 25% of the scores and the last 25% of the scores. Thus it is based only on the central 50% of the scores. • It does not show the scatter from any particular average. • It is not suited to algebraic treatment. 9th - Maths

  32. Dispersion – Quartile Deviation Coefficient of Quartile Deviation It is the ratio between the difference of Quartiles to the sum of quartiles Q1 and Q3 Coefficient of Q .D.= Hence in the above example, = 20 and = 32.6 Coeffecient of Q.D. = 9th - Maths

  33. Dispersion – Mean Deviation • It is also called Average Deviation • Mean Deviation is the average of the deviations of scores from any of the measures of central tendency, the mean, median or mode. • Mean deviation gives best results when deviations are taken from the median. • In averaging deviations to find mean deviation, all deviations whether plus or minus are treated as positive. 9th - Maths

  34. Dispersion – Mean Deviation Calculation of Mean Deviation for Ungrouped data using Median. Calculate the mean deviation for the following data. 10, 20, 23, 12, 16, 19, 22, 13, 27. Solution Step 1: Write the scores in an order 10, 12, 13. 16, 19, 20, 22, 23, 27. Step 2: No of Scores is Odd, i,e, 9 9th - Maths

  35. Dispersion – Mean Deviation Step 3: Find the deviation of each score from the median. Deviation ‘D’ is given by the Score (X - Median). Step 4: Add the Deviations ignoring the '-' sign. This is . denotes only the positive value and is read as Mod D. Step 5: Mean Deviation = where N is the No. of Scores. 9th - Maths

  36. Dispersion – Mean Deviation Calculation of Mean Deviation for Ungrouped data using Mean. Calculate the mean deviation for the following data. 10, 20, 23, 12, 16, 19, 22, 13, 27. Solution Step 1: Write the scores in an order and find the sum of the Scores: 10+12+13+16+19+20+22+23+27=162 Step 2: Mean 9th - Maths

  37. Dispersion – Mean Deviation Step 3: Find the deviation of each score from the mean. Deviation ‘D’ is given by the Score (X - Mean). Step 4: Add the Deviations ignoring the '-' sign. This is . Step 5: Mean Deviation = where N is the No. of Scores. 9th - Maths

  38. Dispersion – Mean Deviation Calculation of Mean Deviation for Grouped data using Median. Calculate the mean deviation for the following data. Step 1: Find the cumulative frequency and compute median for the given data. Step 2: Find the mid point for each class interval (x) and the deviation of each mid point from the median (D) Step 3 : Find the product of the frequency f and deviation D (i.e., ). Then compute 9th - Maths

  39. Dispersion – Mean Deviation Median lies in C.I. 15-19 9th - Maths

  40. Dispersion – Mean Deviation Calculation of Mean Deviation for Grouped data using Mean. Calculate the mean deviation for the following data. Step 1 :Step 1. Find the mid point for each class interval (x) Find the product for each CI Step 2. Find the deviation ‘D’ for each class interval using Step 3. Find the product for each class interval and find 9th - Maths

  41. Dispersion – Mean Deviation 9th - Maths

  42. Graphical Representation A graphic or pictorial treatment can be given to the data which can catch the eye and hold the attention. These graphical representations easily translate numerical facts - often abstract and difficult to interpret - into more concrete and understandable form. Methods • Histogram • Frequency Polygon • Cumulative frequency graph • Cumulative percentage curve 9th - Maths

  43. Graphical Representation - Histogram • Histogram is a graphical representation of a given Frequency Distribution through vertical bars or rectangles. • The combination of all the rectangles gives the histogram. The total area of the histogram represents the total frequency given in the frequency distribution table. 9th - Maths

  44. Graphical Representation - Histogram • Draw histogram for the following data • Step 1: Draw x - axis and y - axis. Mark the origin O. Choose an appropriate scale and convert the frequencies into linear units, Let frequency 1 be represented by 1 cm. • Step 2 : On the x - axis mark the class intervals • Step 3: Draw a rectangle of height 2cm on the first C.l. 0-5 • Step 4: Draw the next rectangle of height 4 cm on the second Cl 5-10. So that two consecutive rectangles have one common side. 9th - Maths

  45. Graphical Representation - Histogram • Final Histogram will be as shown: • Observe that: • the width of each rectangle is the same, because the size of the Class Interval is constant. • there are no gaps between the rectangles which shows that the scores are continuous or not discrete • the height of the rectangles represents frequencies and the base represents the CIs 9th - Maths

  46. Graphical Representation - Histogram • Draw histogram for the following data This distribution is in the inclusive form. Hence this distribution should be first converted to exclusive form For C.I. 5-9 and 10-14: 9th - Maths

  47. Graphical Representation - Histogram • Modified table 9th - Maths

  48. Graphical Representation - Histogram • Final Histogram will be as shown: 9th - Maths

  49. Graphical Representation – Frequency Polygon Frequency polygon is another graphical representation of a frequency distribution. This can be drawn in two ways. • By drawing a Histogram for the given frequency distribution and then frequency polygon. • By marking the mid points of Class Intervals on x - axis and respective frequencies on y - axis and plotting the points. 9th - Maths

  50. Graphical Representation – Frequency Polygon Method 1: Step 1: Construct the histogram Scale: x-axis: I cm = size of C.I.; y-axis I cm=1f Step 2: Mark the mid-points of the tops of rectangles of histogram. Join the mid-points by line segments Step 3: To complete the polygon, mark two imaginary classes -5-0 and 35-40. Mark their mid points. Step 4: Join the end points of the line segments to the midpoints of the two imaginary classes. We get the required frequency polygon. 9th - Maths

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