Successfully reported this slideshow.
We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime.
Upcoming SlideShare
What to Upload to SlideShare
What to Upload to SlideShare
Loading in …3
×
1 of 58

measure of dispersion

31

Share

Download to read offline

examples
normal desertion

Related Books

Free with a 30 day trial from Scribd

See all

Related Audiobooks

Free with a 30 day trial from Scribd

See all

measure of dispersion

  1. 1. Case 6: BIOSTATISTICS “Measures of Dispersion”. G5-PBL
  2. 2. Objectives: Types of data. Measures of Dispersion. Range. Quartile deviation. Mean deviation. Standard deviation. Suitability of a dispersion method. Case solving.
  3. 3. Types of data 1- Qualitative Data 2- Quantitative Data
  4. 4. 1- Qualitative Data ● Uses words and descriptions. ● Can be observed. ● Examples of qualitative data: descriptions of texture, taste, or an experience.
  5. 5. 2- Quantitative Data ● Expressed with numbers. ● Can be put into categories, measured, or ranked. ● Examples of quantitative data: length, weight, age, cost, salary.
  6. 6. The Quantitative Data has two types: 1- Categorical data 2- Continuous data - has been placed into groups. - Example: hair color, opinions. - numerical data measured on a continuous range or scale. - Example: height, weight.
  7. 7. What might be the qualitative and quantitative data that describe this cup of coffee? - The qualitative data: it has a strong taste and robust aroma. - The quantitative data: it is 150 degrees Fahrenheit, and costs 10 SR.
  8. 8. MEASURES OF DISPERSION Introduction In addition to the measures of central tendency such as mean, mode,median we often need to calculate a second type of measure called a measure of dispersion which measures the variation in the observations about the middle value– mean or median etc. ● The measure of central tendency of any series or data distribution summarises it into single representative for which are useful in many respect but it fails to account the general distribution pattern of data. ● Thus any conclusion only based on central tendency may be misleading ● Dispersion can prove very effective in association with central tendency in making any statistical decision.
  9. 9. WHAT IS MEASURES OF DISPERSION ? Measures of dispersion: group of analytical tools that describes the spread or variability of a data set Suppose that we have the distribution of the yields (kg per plot) of two paddy varieties from 5 plots each. The distribution may be as follows: Variety I 45 42 42 41 40 Variety II 54 48 42 33 30 It can be seen that the mean yield for both varieties is 42 kg but cannot say that the performances of the two varieties are same. There is greater uniformity of yields in the first variety whereas there is more variability in the yields of the second variety. The first variety may be preferred since it is more consistent in yield performance.
  10. 10. It is the value of dispersion which says how much reliable a central tendency is? Usually, a small value of dispersion indicates that measure of central tendency is more reliable representative of data series and vice‐versa. There are different measures of dispersion like the range, the quartile deviation, the mean deviation and the standard deviation IMPORTANCE OF MEASURES OF DISPERSION ● supplements an average or a measure of central tendency. ● compares one group of data with another. ● Indicates how representative the data is. ● Measure of dispersion is also used to compare uniformity of different data like income, temperature, rainfall, weight, height… etc.
  11. 11. The range in statistics is the difference between the maximum and minimum values of a data set. We will learn more details concerning this very basic descriptive statistic
  12. 12. Example: ● (8 , 9, 10 , 11 , 12) Range: 12-8=4 ● (0 , 10 , -10 , 30 ,20) (-10 , 0 , 10 , 20 ,30) Range: 30-(-10)=40
  13. 13. Range for Ungrouped data. *Range: 3.1 - 0.5 = 2.6 Time Frequency 0.5 4 1.3 5 1.6 3 2.3 9 2.6 1 3.1 3
  14. 14. Range for Grouped data. *Range: 5 - 1.5 = 3.5 Time Frequency (Midpoint) X 1.2 - 1.8 10 1.5 1.9 - 2.5 11 2.2 2.6 - 3.2 5 2.9 3.3 - 3.9 3 3.6 4.0 - 4.6 2 4.3 4.7 - 5.3 1 5
  15. 15. Example: The answer is: c
  16. 16. The answer is: c
  17. 17. Uses of Range With all its limitations Range is commonly used in certain fields. These are: (i) Quality Control: In quality control of manufactured products, range is used to study the variation in the quality of the units manufactured. Even with the most modern mechanical equipment there may be a small, almost insignificant, difference in the different units of a commodity manufactured. Thus, if a company is manufacturing bottles of a particular type, there may be a slight variation in the size or shape of the bottles manufactured. In such cases a range is usually determined, and all the units which fall within these limits are accepted while those which fall outside the limits are rejected.
  18. 18. Uses of Range (ii) Variation in Money Rates, Share values, Exchange Rates and Gold prices, etc: Variations in money rates, share values, gold prices and exchange rates are commonly studied through range because the fluctuations in them are not very large. In fact range as a measure of dispersion should be generally used only when variations in the value of the variable are not much. (iii) Weather forecasting: Range gives an idea of the variation between maximum and minimum levels of temperature. From day to day the range would not vary much and it is helpful in studying the vagaries of nature if variations suddenly rise or fall.
  19. 19. Quartile Deviation Quartiles in statistics are values that divide your data into quarters. they divide your data into four segments according to where the numbers fall on the number line. The four quarters that divide a data set into quartiles are: 1. The lowest 25% of numbers. 2. The next lowest 25% of numbers (up to the median). 3. The second highest 25% of numbers (above the median). 4. The highest 25% of numbers.
  20. 20. How can you find the quartile ? If a data set of scores is arranged in ascending order of magnitude, then: The lower quartile (Q1) is the median of the lower half of the data set. Q 2 (The median) is the middle value of the data set The upper quartile (Q3) is the median of the upper half of the data set.
  21. 21. Quartile Deviation semi-inter-quartile range or the quartile deviation: Quartile Deviation (QD) means the semi variation between the upper quartiles (Q3) and lower quartiles (Q1) in a distribution the inter quartile rang (IQR) : is the spread of the middle 50% of the data values. So: Coefficient of Quartile Deviation:A relative measure of dispersion based on the quartile deviation is called the coefficient of quartile deviation.
  22. 22. Quartile Deviation For Ungrouped Data First: Arrange the data in ascending order. Second: Find First Quartile Third: Find Third Quartile Finally: Put the values into the Formula of Quartile Deviation
  23. 23. Example: Problem: Following are run scores by batsman in last 20 test matches: 96,70,100,96,81,84,90,89,63,90,34,75,39,82,85,86,76,64,67 and 88.
  24. 24. First: Arrange the data in Ascending order 34,39,63,64,67,70,75,76,81,82,84,85,86,8 8,89,90,90,96,96,100
  25. 25. Second: First Quartile
  26. 26. Third: Third Quartile
  27. 27. Finally: find Quartile Deviation
  28. 28. Quartile Deviation For grouped Data Calculate the quartile deviation and coefficient of quartile deviation from the data given below:
  29. 29. Quartile Deviation For grouped Data We have to calculate: ● Class boundaries : The lower limit for every class is the smallest value in that class. the upper limit for every class is the greatest value in that class. The size of the gap between classes is the difference between the upper class limit of one class and the lower class limit of the next class. In this case the gap is 9.8-9.7 = 0.1 The lower boundary of each class is calculated by subtracting half of the gap value 9.3 - (0.1 / 2 ) =9.25 the upper boundary of each class is calculated by adding half of the gap value 9.7 + (0.1 /2 ) = 9.75
  30. 30. Quartile Deviation For grouped Data We have to calculate: ● Cumulative frequency The total of a frequency and all frequencies so far in a frequency distribution. It is the 'running total' of frequencies.
  31. 31. Quartile Deviation For grouped Data
  32. 32. Mean Deviation. Mean Deviation is the arithmetic mean of the differences of the values from their average. The average used is either the arithmetic mean or median. Since the average is a central value, some deviations are positive and some are negative. If these are added as they are, the sum will not reveal anything; because the sum of deviations from Arithmetic Mean is always zero. Mean Deviation tries to overcome this problem by ignoring the signs of deviations, i.e., it considers all deviations positive.
  33. 33. Mean Deviation for ungrouped data. Direct Method Steps: The A.M. of the values is calculated. Difference between each value and the A.M. is calculated. All differences are considered positive. These are denoted as |d| The A.M. of these differences (called deviations) is the Mean Deviation. i.e. MD = Σ | d | n
  34. 34. Example. Calculate the Mean Deviation of the following values; 2, 4, 7, 8 and 9.
  35. 35. Solution. 1.
  36. 36. Mean Deviation for Grouped Data Steps 1. Calculate the mean of the distribution. 2. Calculate the absolute deviations |d| of the class midpoints from the mean. 3. Multiply each |d| value with its corresponding frequency to get f|d| values. Sum them up to get ∑f∣d∣. 4. Apply the following formula, M.D. = ∑f∣d∣ ∑f
  37. 37. Example Calculate the mean deviation of the following distribution: Profits of Companies (Rs in lakhs) Number of Companies 11-20 5 21-30 8 31-40 16 41-50 8 51-60 3
  38. 38. Solution M.D. = ∑f|d| = 334 = 8.35 ∑f 40 Class Intervals Frequency Mid-point (x) |d| f|d| 11-20 5 15.5 19 95 21-30 8 25.5 9 72 31-40 16 35.5 1 16 41-50 8 45.5 11 88 51-60 3 55.5 21 63 ∑f=40 ∑f|d|=334
  39. 39. Standard deviation The Standard Deviation is a number that measures how far away each number in a set of data is from their mean ,it shows the variation in data. Standard Deviation is also known as root-mean square deviation as it is the square root of means of the squared deviations from the arithmetic mean. If the Standard Deviation is large it means the numbers are spread out from their mean. If the Standard Deviation is small it means the numbers are close to their mean. The symbol for Standard Deviation used for a sample data is s and the symbol used for a population data is σ.
  40. 40. Formulas of Standard Deviation For ungrouped In case of individual observations, Standard Deviation can be computed in any of the two ways: 1. Take the deviation of the items from the actual mean 2. Take the deviation of the item from the assumed mean The formula for an ungroup data is: ´The formula for an ungroup data using assumed mean method: where d=x-a
  41. 41. Example Step1 take the square of the values Step 2 take square of each value Step 3 take sum of both columns Step 4 Apply them to the formula S= ⎷(83/3 - (15)2/9) S= 1.41 x x 2 3 9 5 25 7 49 Σ=15 Σ=83
  42. 42. Where Standard deviation is Applied Investment firms use Standard deviation for their mutual funds. In financial terms, standard deviation is used to measure risk involved in an investment. In physical experiments, it is important to have a measurement of uncertainty. Standard deviation provides a way to check the results. Very large values of standard deviation can mean the experiment is faulty. Web Analytics will help you get an idea of just how important events that happen on your website could be using Standard deviation.
  43. 43. grouped data standard Deviation M = mid-point μ = Mean F = frequency n= number of samples Σ=sum σ2 = data variance σ= standard deviation
  44. 44. Example Problem: Find an estimate of the variance and standard deviation of the following data for the marks obtained in a test by 88 students. Marks Frequency (f) 0 ≤ x < 10 6 10 ≤ x < 20 16 20 ≤ x < 30 24 30 ≤ x < 40 25 40 ≤ x < 50 17
  45. 45. step 1:find the mid-point for each group or range of the frequency table. (0+10)/2 = 5 (10+20)/2= 15 (20+30)/2= 25 (30+40)/2= 35 (40+50)/2= 45
  46. 46. step 2: calculate the number of samples of a data set by summing up the frequencies. n= 6+16+24+25+17= 88 step 3: find the mean for the grouped data by dividing the addition of multiplication of each group mid-point and frequency of the data set by the number of samples. μ= Σ(M*F)/n μ= (5×6 + 15×16 + 25×24 + 35×25 + 45×17)/n μ= 2510/88 = 28.5227
  47. 47. Step 4: calculate the variance for the frequency table data. σ2=Σ(F × M2) - (n × μ2)/(n - 1) σ2= (6×52 + 16×152 + 24×252 + 25×352 + 17×452) - (88 × (28.5227)2)/(88-1) σ2= 138.73 step 5:estimate standard deviation for the frequency table by taking square root of the variance. σ= √138.73 = 11.78
  48. 48. Solve the case Find the following measures: 1. Range 2. Mean deviation 3. Standard deviation 4. Quartile deviation Time Frequency 0.5 4 1.3 5 1.6 3 2.3 9 2.6 1 3.1 3 Time Frequency 1.2-1.8 10 1.9-2.5 11 2.6-3.2 5 3.3-3.9 3 4.0-4.6 2 4.7-5.3 1
  49. 49. Use the following formulas to solve the problem 1. Range=maximum−minimum 2. Quantile deviation=Q3-Q1/2 Grouped Q1= L+((N/4*CF)/F)i Q3=L+((3n-CF/F))i Ungrouped Q1=(n+1)/4 Q3=3(n+1)/4
  50. 50. 3.Mean deviation Grouped=√((∑fd2/n)-(∑fd)2/n) Ungrouped=∑(d)/n 4.standard deviation Grouped=√((∑fd2)/n-(∑fd)2/n) Ungrouped=√(∑d2)/n
  51. 51. time frequ ency m.p c.f fx d d2 fd 1.2- 1.8 10 1.5 10 15 276 7617 6 2760 1.9- 2.5 11 4.4 21 48.4 242.6 5885 4.76 2668. 6 2.6- 3.2 5 5.8 26 29 262 6864 4 1310 3.3- 3.9 3 7.2 29 21.6 269.4 7257 6.36 808.2 4.0- 4.6 2 8.6 31 172 119 1416 1 238 4.7- 5.3 1 5 32 5 286 8179 6 286
  52. 52. tme frequency c.f d d2 fd fd2 0.5 4 4 3.6 12.96 14.4 51.84 1.3 5 11 2.8 7.84 14 39.2 1.6 3 14 2.5 6.25 7.5 18.75 2.3 9 23 1.86 3.46 16.74 31.14 2.6 1 24 1.56 2.43 1.56 1.56 3.1 3 27 1.06 1.12 3.18 9.54
  53. 53. Suitability of a dispersion method Range is the simplest method of studying dispersion. Range is the difference between the smallest value and the largest value of a series. While computing range, we do not take into account frequencies of different groups. Formula: Absolute Range = L – S Coefficient of Range = where, L represents largest value in a distribution S represents smallest value in a distribution
  54. 54. Standard Deviation Advantages:- •Shows how much data is clustered around a mean value •It gives a more accurate idea of how the data is distributed •Not as affected by extreme values Disadvantages:- •It doesn't give you the full range of the data •It can be hard to calculate •Only used with data where an independent variable is plotted against the frequency of it •Assumes a normal distribution pattern
  55. 55. The mean deviation is actually more efficient than the standard deviation in the realistic situation where some of the measurements are in error, more efficient for distributions other than perfect normal, closely related to a number of other useful analytical techniques, and easier to understand.
  56. 56. References ● https://www.thoughtco.com/what-is-the-range-in-statistics-3126248 ● http://www.transtutors.com/homework-help/statistics/measures-of-dispersion/uses-range.aspx ● http://www.mathopolis.com/questions/q.php?id=696&site=1&ref=/data/range.html&qs=696_740_1468_1469_2159_2160_3064_3065_3 798_3799 ● http://www.mathopolis.com/questions/q.php?id=3799&site=1&ref=/data/range.html&qs=696_740_1468_1469_2159_2160_3064_3065_ 3798_3799 ● https://www.youtube.com/watch?v=vuhB1O8q0ys ● http://mba-lectures.com/statistics/descriptive-statistics/279/range.html ● http://www.kuk.ac.in/userfiles/file/distance_education/Year-2011-2012/Lecture-2%20(Paper%205(a)).pdf ● http://eagri.tnau.ac.in/eagri50/STAM101/pdf/lec05.pdf ● http://ncert.nic.in/ncerts/l/keep215.pdf ● http://labstat.psa.gov.ph/learnstat/3_Measures%20of%20Central%20Tendency,%20Dispersion%20and%20Skewness.pdf ● http://www.emathzone.com/tutorials/basic-statistics/quartile-deviation-and-its-coefficient.html ● http://www.mathsteacher.com.au/year10/ch16_statistics/05_quartiles/24quartiles.htm ● http://www.statisticshowto.com/what-are-quartiles/ ● https://www.easycalculation.com/statistics/learn-quartile-deviation-calculator.php ● https://www.mathway.com/examples/statistics/frequency-distribution/finding-the-class-boundaries?id=1003
  57. 57. HAVE A NICE DAY :)

Editor's Notes

  • Hedaya
  • Hedaya
  • Hedaya
  • Hedaya
  • Hedaya
  • Hedaya
  • KAT
    KHADIJA
  • Khadijah

    Form the above example it is obvious that a measure of central tendency alone is not sufficient to describe a frequency distribution. In addition to it we should have a measure of scatterness of observations. The scatterness or variation of observations from their average are called the dispersion.

  • KAT
    KHADIJA
  • Yasmen Tawfeeq Rasheed
    110110
  • Yasmen Tawfeeq Rasheed
    110110
  • Yasmen Tawfeeq Rasheed
    110110
  • Yasmen Tawfeeq Rasheed
    110110
  • salma
  • salma
  • salma
  • salma
  • Mariam
  • mariam
  • mariam
  • hoda
  • hoda
  • hoda
  • hoda
  • hoda
  • hoda
  • mariam
  • mariam
  • mariam
  • mariam
  • kainat
  • kainat
  • kainat
  • kainat
  • kawthar
  • kawthar
  • kawthar
  • kawthar
  • kawthar
  • Maria and hafsat
  • Maria and hafsat
  • Maria and hafsat
  • Baraa
  • Rowaida Ahmad Alwalidi 110305
    Slides number 54-55
  • Rowaida Ahmad Alwalidi 110305
    Slides number 54-55
  • ×