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### Introduction to Statistics and Statistical Inference

• 1. Note: Most of the Slides were taken from Elementary Statistics: A Handbook of Slide Presentation prepared by Z.V.J. Albacea, C.E. Reano, R.V. Collado, L.N. Comia and N.A. Tandang in 2005 for the Institute of Statistics, CAS, UP Los Banos Training on Teaching Basic Statistics for Tertiary Level Teachers Summer 2008 INTRODUCTION TO STATISTICS AND STATISTICAL INFERENCE
• 2. Session 1.2 TEACHING BASIC STATISTICS …. Florence Nightingale on Statistics  “...the most important science in the whole world: for upon it depends the practical application of every other science and of every art: the one science essential to all political and social administration, all education, all organization based on experience, for it only gives results of our experience.”  “To understand God's thoughts, we must study statistics, for these are the measures of His purpose.”
• 3. Session 1.3 TEACHING BASIC STATISTICS …. Realities about Statistics  The man in the street distrusts statistics and despises [his image of] statisticians, those who diligently collect irrelevant facts and figures and use them to manipulate society. “There are three kinds of lies: lies, damned lies, and statistics” – Mark Twaine  One can not go about without statistics. “Statistics are like bikinis. What they reveal is suggestive, but what they conceal is vital.” – Aaron Levenstein
• 4. Session 1.4 TEACHING BASIC STATISTICS …. Definition of Statistics plural sense: numerical facts, e.g. CPI, peso-dollar exchange rate singular sense: scientific discipline consisting of theory and methods for processing numerical information that one can use when making decisions in the face of uncertainty.
• 5. Session 1.5 TEACHING BASIC STATISTICS …. History of Statistics  The term statistics came from the Latin phrase “ratio status” which means study of practical politics or the statesman’s art.  In the middle of 18th century, the term statistik (a term due to Achenwall) was used, a German term defined as “the political science of several countries”  From statistik it became statistics defined as a statement in figures and facts of the present condition of a state.
• 6. Session 1.6 TEACHING BASIC STATISTICS …. Application of Statistics  Diverse applications “During the 20th Century statistical thinking and methodology have become the scientific framework for literally dozens of fields including education, agriculture, economics, biology, and medicine, and with increasing influence recently on the hard sciences such as astronomy, geology, and physics. In other words, we have grown from a small obscure field into a big obscure field.” – Brad Efron
• 7. Session 1.7 TEACHING BASIC STATISTICS …. Application of Statistics  Comparing the effects of five kinds of fertilizers on the yield of a particular variety of corn  Determining the income distribution of Filipino families  Comparing the effectiveness of two diet programs  Prediction of daily temperatures  Evaluation of student performance
• 8. Session 1.8 TEACHING BASIC STATISTICS …. Two Aims of Statistics Statistics aims to uncover structure in data, to explain variation…  Descriptive  Inferential
• 9. Session 1.9 TEACHING BASIC STATISTICS …. Areas of Statistics Descriptive statistics  methods concerned w/ collecting, describing, and analyzing a set of data without drawing conclusions (or inferences) about a large group Inferential statistics  methods concerned with the analysis of a subset of data leading to predictions or inferences about the entire set of data
• 10. Session 1.10 TEACHING BASIC STATISTICS …. Example of Descriptive Statistics Present the Philippine population by constructing a graph indicating the total number of Filipinos counted during the last census by age group and sex
• 11. Session 1.11 TEACHING BASIC STATISTICS …. Example of Inferential Statistics A new milk formulation designed to improve the psychomotor development of infants was tested on randomly selected infants. Based on the results, it was concluded that the new milk formulation is effective in improving the psychomotor development of infants.
• 12. Session 1.12 TEACHING BASIC STATISTICS …. Inferential Statistics Larger Set (N units/observations) Smaller Set (n units/observations) Inferences and Generalizations
• 13. Session 1.13 TEACHING BASIC STATISTICS …. Key Definitions  The universe/physical population is the collection of things or observational units under consideration.  A variable is a characteristic observed or measured on every unit of the universe.  The statistical population is the set of all possible values of the variable.  Measurement is the process of determining the value or label of the variable based on what has been observed.  An observation is the realized value of the variable.  Data is the collection of all observations.
• 14. Session 1.14 TEACHING BASIC STATISTICS …. Key Definitions  Parameters are numerical measures that describe the population or universe of interest. Usually donated by Greek letters; µ (mu), σ (sigma), ρ (rho), λ (lambda), τ (tau), θ (theta), α (alpha) and β (beta).  Statistics are numerical measures of a sample
• 15. Session 1.15 TEACHING BASIC STATISTICS …. VARIABLES Qualitative Quantitative ContinuousDiscrete Types of Variables Qualitative variable  Describes the quality or character of something Quantitative variable  Describes the amount or number of something a. Discrete  countable a. Continuous  Measurable (measured using a continuous scale such as kilos, cms, grams) a. Constant
• 16. Session 1.16 TEACHING BASIC STATISTICS …. Levels of Measurement 1. Nominal  Numbers or symbols used to classify units into distinct categories 1. Ordinal scale  Accounts for order; no indication of distance between positions 1. Interval scale  Equal intervals (fixed unit of measurement); no absolute zero 1. Ratio scale  Has absolute zero
• 17. Session 1.17 TEACHING BASIC STATISTICS …. Methods of Collecting Data  Objective Method  Subjective Method  Use of Existing Records
• 18. Session 1.18 TEACHING BASIC STATISTICS …. Methods of Presenting Data Textual Tabular Graphical
• 19. Session 1.19 TEACHING BASIC STATISTICS …. Mean Median Mode Summary Measures Variation Variance Standard Deviation Coefficient of Variation Range Location Maximum Minimum Percentile Quartile Decile Median Interquartile Range Skewness Kurtosis Central Tendency
• 20. Session 1.20 TEACHING BASIC STATISTICS ….  A single value that is used to identify the “center” of the data it is thought of as a typical value of the distribution precise yet simple most representative value of the data Measures of Central Tendency
• 21. Session 1.21 TEACHING BASIC STATISTICS …. Mean  Most common measure of the center  Also known as arithmetic average 1 1 2 N i i N X X X X N N µ = + + + = = ∑ K 1 21 n i ni x x x x x n n = + + + = = ∑ K Population Mean: Sample Mean:
• 22. Session 1.22 TEACHING BASIC STATISTICS …. Properties of the Mean  may not be an actual observation in the data set  can be applied in at least interval level  easy to compute  every observation contributes to the value of the mean
• 23. Session 1.23 TEACHING BASIC STATISTICS …. Properties of the Mean  subgroup means can be combined to come up with a group mean (use weighted mean)  easily affected by extreme values 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14 Mean = 5 Mean = 6
• 24. Session 1.24 TEACHING BASIC STATISTICS …. Median  Divides the observations into two equal parts  If the number of observations is odd, the median is the middle number.  If the number of observations is even, the median is the average of the 2 middle numbers.  Sample median denoted as while population median is denoted as x~ µ~
• 25. Session 1.25 TEACHING BASIC STATISTICS …. Properties of a Median  may not be an actual observation in the data set  can be applied in at least ordinal level  a positional measure; not affected by extreme values 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14 Median = 5
• 26. Session 1.26 TEACHING BASIC STATISTICS …. Mode  occurs most frequently  nominal average  may or may not exist 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Mode = 9 0 1 2 3 4 5 6 No Mode
• 27. Session 1.27 TEACHING BASIC STATISTICS …. Properties of a Mode  can be used for qualitative as well as quantitative data  may not be unique  not affected by extreme values  can be computed for ungrouped and grouped data
• 28. Session 1.28 TEACHING BASIC STATISTICS …. Mean, Median & Mode Use the mean when:  sampling stability is desired  other measures are to be computed
• 29. Session 1.29 TEACHING BASIC STATISTICS …. Mean, Median & Mode Use the median when:  the exact midpoint of the distribution is desired  there are extreme observations
• 30. Session 1.30 TEACHING BASIC STATISTICS …. Mean, Median & Mode Use the mode when:  when the "typical" value is desired  when the dataset is measured on a nominal scale
• 31. Session 1.31 TEACHING BASIC STATISTICS …. Measures of Location  A Measure of Location summarizes a data set by giving a value within the range of the data values that describes its location relative to the entire data set arranged according to magnitude (called an array). Some Common Measures:  Minimum, Maximum  Percentiles, Deciles, Quartiles
• 32. Session 1.32 TEACHING BASIC STATISTICS …. Maximum and Minimum  Minimum is the smallest value in the data set, denoted as MIN.  Maximum is the largest value in the data set, denoted as MAX.
• 33. Session 1.33 TEACHING BASIC STATISTICS …. Percentiles  Numerical measures that give the relative position of a data value relative to the entire data set.  Divide an array (raw data arranged in increasing or decreasing order of magnitude) into 100 equal parts.  The jth percentile, denoted as Pj, is the data value in the the data set that separates the bottom j% of the data from the top (100-j)%.
• 34. Session 1.34 TEACHING BASIC STATISTICS …. EXAMPLE Suppose LJ was told that relative to the other scores on a certain test, his score was the 95th percentile.  This means that (at least) 95% of those who took the test had scores less than or equal to LJ’s score, while (at least) 5% had scores higher than LJ’s.
• 35. Session 1.35 TEACHING BASIC STATISTICS …. Deciles  Divide an array into ten equal parts, each part having ten percent of the distribution of the data values, denoted by Dj.  The 1st decile is the 10th percentile; the 2nd decile is the 20th percentile…..
• 36. Session 1.36 TEACHING BASIC STATISTICS …. Quartiles  Divide an array into four equal parts, each part having 25% of the distribution of the data values, denoted by Qj.  The 1st quartile is the 25th percentile; the 2nd quartile is the 50th percentile, also the median and the 3rd quartile is the 75th percentile.
• 37. Session 1.37 TEACHING BASIC STATISTICS …. Measures of Variation  A measure of variation is a single value that is used to describe the spread of the distribution A measure of central tendency alone does not uniquely describe a distribution
• 38. Session 1.38 TEACHING BASIC STATISTICS …. Mean = 15.5 s = 3.33811 12 13 14 15 16 17 18 19 20 21 11 12 13 14 15 16 17 18 19 20 21 Data B Data A Mean = 15.5 s = .9258 11 12 13 14 15 16 17 18 19 20 21 Mean = 15.5 s = 4.57 Data C A look at dispersion…
• 39. Session 1.39 TEACHING BASIC STATISTICS …. Two Types of Measures of Dispersion Absolute Measures of Dispersion:  Range  Inter-quartile Range  Variance  Standard Deviation Relative Measure of Dispersion:  Coefficient of Variation
• 40. Session 1.40 TEACHING BASIC STATISTICS …. Range (R) The difference between the maximum and minimum value in a data set, i.e. R = MAX – MIN Example: Pulse rates of 15 male residents of a certain village 54 58 58 60 62 65 66 71 74 75 77 78 80 82 85 R = 85 - 54 = 31
• 41. Session 1.41 TEACHING BASIC STATISTICS …. Some Properties of the Range  The larger the value of the range, the more dispersed the observations are.  It is quick and easy to understand.  A rough measure of dispersion.
• 42. Session 1.42 TEACHING BASIC STATISTICS …. Inter-Quartile Range (IQR) The difference between the third quartile and first quartile, i.e. IQR = Q3 – Q1 Example: Pulse rates of 15 residents of a certain village 54 58 58 60 62 65 66 71 74 75 77 78 80 82 85 IQR = 78 - 60 = 18
• 43. Session 1.43 TEACHING BASIC STATISTICS …. Some Properties of IQR  Reduces the influence of extreme values.  Not as easy to calculate as the Range.
• 44. Session 1.44 TEACHING BASIC STATISTICS …. Variance  important measure of variation  shows variation about the mean Population variance Sample variance N X N i i∑= − = 1 2 2 )( µ σ 1 )( 1 2 2 − − = ∑= n xx s n i i
• 45. Session 1.45 TEACHING BASIC STATISTICS …. Standard Deviation (SD)  most important measure of variation  square root of Variance  has the same units as the original data Population SD Sample SD N X N i i∑= − = 1 2 )( µ σ 1 )( 1 2 − − = ∑= n xx s n i i
• 46. Session 1.46 TEACHING BASIC STATISTICS …. (Sample) Data: 10 12 14 15 17 18 18 24 n = 8 Mean =16 309.4 7 2)1624(2)1618(2)1617(2)1615(2)1614(2)1612(2)1610( = −+−+−+−+−+−+− =s Computation of Standard Deviation
• 47. Session 1.47 TEACHING BASIC STATISTICS …. Remarks on Standard Deviation  If there is a large amount of variation, then on average, the data values will be far from the mean. Hence, the SD will be large.  If there is only a small amount of variation, then on average, the data values will be close to the mean. Hence, the SD will be small.
• 48. Session 1.48 TEACHING BASIC STATISTICS …. Mean = 15.5 s = 3.33811 12 13 14 15 16 17 18 19 20 21 11 12 13 14 15 16 17 18 19 20 21 Data B Data A Mean = 15.5 s = .9258 11 12 13 14 15 16 17 18 19 20 21 Mean = 15.5 s = 4.57 Data C Comparing Standard Deviations (comparable only when units of measure are the same and the means are not too different from each other)
• 49. Session 1.49 TEACHING BASIC STATISTICS …. Example: Team A - Heights of five marathon players in inches 5” 65 “ 65 “ 65 “ 65 “ 65 “ Mean = 65 S = 0 Comparing Standard Deviations
• 50. Session 1.50 TEACHING BASIC STATISTICS …. Example: Team B - Heights of five marathon players in inches 62 “ 67 “ 66 “ 70 “ 60 “ Mean = 65” s = 4.0” Comparing Standard Deviation
• 51. Session 1.51 TEACHING BASIC STATISTICS …. Properties of Standard Deviation  It is the most widely used measure of dispersion. (Chebychev’s Inequality)  It is based on all the items and is rigidly defined.  It is used to test the reliability of measures calculated from samples.  The standard deviation is sensitive to the presence of extreme values.  It is not easy to calculate by hand (unlike the range).
• 52. Session 1.52 TEACHING BASIC STATISTICS …. Chebyshev’s Rule It permits us to make statements about the percentage of observations that must be within a specified number of standard deviation from the mean The proportion of any distribution that lies within k standard deviations of the mean is at least 1-(1/k2 ) where k is any positive number larger than 1. This rule applies to any distribution.
• 53. Session 1.53 TEACHING BASIC STATISTICS …. For any data set with mean (µ) and standard deviation (SD), the following statements apply: At least 75% of the observations are within 2SD of its mean. At least 88.9% of the observations are within 3SD of its mean. Chebyshev’s Rule
• 54. Session 1.54 TEACHING BASIC STATISTICS …. Illustration At least 75% At least 75% of the observations are within 2SD of its mean.
• 55. Session 1.55 TEACHING BASIC STATISTICS …. Example The midterm exam scores of 100 STAT 1 students last semester had a mean of 65 and a standard deviation of 8 points. Applying the Chebyshev’s Rule, we can say that: 1. At least 75% of the students had scores between 49 and 81. 2. At least 88.9% of the students had scores between 41 and 89.
• 56. Session 1.56 TEACHING BASIC STATISTICS …. Coefficient of Variation (CV)  measure of relative variation  usually expressed in percent  shows variation relative to mean  used to compare 2 or more groups  Formula : 100%×      = Mean SD CV
• 57. Session 1.57 TEACHING BASIC STATISTICS …. Comparing CVs  Stock A: Average Price = P50 SD = P5 CV = 10%  Stock B: Average Price = P100 SD = P5 CV = 5%
• 58. Session 1.58 TEACHING BASIC STATISTICS …. Measure of Skewness  Describes the degree of departures of the distribution of the data from symmetry.  The degree of skewness is measured by the coefficient of skewness, denoted as SK and computed as, ( ) SD MedianMean K − = 3 S
• 59. Session 1.59 TEACHING BASIC STATISTICS …. What is Symmetry? A distribution is said to be symmetric about the mean, if the distribution to the left of mean is the “mirror image” of the distribution to the right of the mean. Likewise, a symmetric distribution has SK=0 since its mean is equal to its median and its mode.
• 60. Session 1.60 TEACHING BASIC STATISTICS …. SK > 0 positively skewed Measure of Skewness SK < 0 negatively skewed
• 61. Session 1.61 TEACHING BASIC STATISTICS …. Measure of Kurtosis  Describes the extent of peakedness or flatness of the distribution of the data.  Measured by coefficient of kurtosis (K) computed as, ( )4 1 4 3 N i i X K N µ σ = − = − ∑
• 62. Session 1.62 TEACHING BASIC STATISTICS …. K = 0 mesokurtic K > 0 leptokurtic K < 0 platykurtic Measure of Kurtosis
• 63. Session 1.63 TEACHING BASIC STATISTICS …. Box-and-Whiskers Plot  Concerned with the symmetry of the distribution and incorporates measures of location in order to study the variability of the observations.  Also called as box plot or 5-number summary (represented by Min, Max, Q1, Q2, and Q3).  Suitable for identifying outliers.
• 64. Session 1.64 TEACHING BASIC STATISTICS …. The diagram is made up of a box which lies between the first and third quartiles. The whiskers are the straight lines extending from the ends of the box to the smallest and largest values that are not outliers. Box-and-Whiskers Plot
• 65. Session 1.65 TEACHING BASIC STATISTICS …. Steps to Construct a Box-and-Whiskers plot Step 1: Draw a rectangular box whose left edge is at the Q1 and whose right edge is at the Q3 so the box width is the IQR. Then draw a vertical line segment inside the box where the median is found. Q1 Q3Md 75 78 85
• 66. Session 1.66 TEACHING BASIC STATISTICS …. Step 2: Place marks at distances 1.5 IQR from either end of the box. (1.5 IQR =15) 100 Q1 Q3Md 75 78 8560 1.5 IQR 1.5 IQR Steps to Construct a Box-and-Whiskers plot
• 67. Session 1.67 TEACHING BASIC STATISTICS …. Step 3:Draw the horizontal line segments known as the “whiskers” from each of the end box to the largest and smallest values in the data set that are not outliers. (An observation beyond ±1.5 IQR is an outlier.) Steps to Construct a Box-and-Whiskers plot
• 68. Session 1.68 TEACHING BASIC STATISTICS …. Step 4: For every outlier, draw a dot. If two or more dots have the same values, draw the dots side by side. Q1 Q3 Md 75 78 8560 100 1.5 IQR 1.5 IQR 9855 . . Steps to Construct a Box-and-Whiskers plot
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