Elementary Statistics

Transcription

1 Elementary Statistics Dr. Ghamsary Chapter Page 1 Elementary Statistics M. Ghamsary, Ph.D. Chapter 0 1

2 Elementary Statistics Dr. Ghamsary Chapter Page Descriptive Statistics Grouped vs Ungrouped Data Ungrouped data: have not been summarized in any way are also called raw data Grouped data: have been organized into a frequency distribution Raw Data: When data are collected in original form, they are called raw data. The following are the scores on the first test of the statistics class in fall of Table.1: Data fromtest#1 of fall 007 Stem-and-Leaf: One method of displaying a set of data is with a stem-and-leaf plot. Stem Leaf Group Data: When the raw data is organized into a frequency distribution Frequency Distribution: is the organizing of raw data in table form, using classes and frequencies.

3 Elementary Statistics Dr. Ghamsary Chapter Page 3 Class Tally Frequency Class: Number of classes in the above table is 5. Class Limits: represent the smallest and largest data values in each class. Lower Class: the lowest number in each class. In above table 50 is the lower class limit of the first class, 60 is the lower class limit of the nd class, etc. Upper Class: the highest number in each class. In above table 59 is the upper class limit of the first class, 69 is the upper class limit of the nd class, etc. Class Width: for a class in a frequency distribution is found by subtracting the lower (or upper) class limit of one class minus the lower (or upper) class limit of the previous class. In above table the class width is 10. Class Boundaries are used to separate the classes so that there are no gaps in the frequency distribution. Class Class Frequency Boundaries

4 Elementary Statistics Dr. Ghamsary Chapter Page 4 Cumulative Frequency: Relative Frequency: Class Frequency Cumulative Frequency Relative Frequency /50= =15 9/50= =7 1/50= =4 15/50= =50 8/50=0.16 n=50 Most Popular Graphs in Statistics The most commonly used graphs in statistics are: 1. The Histogram. The Frequency Polygon. 3. The Cumulative Frequency Graph 4. The Bar Chart 5. Pie Chart 6. Pareto Charts 7. Dot Plot 8. Stem-Leaf 9. Time Series Graph 1. The Histogram o Making decisions about a process, product, or procedure that could be improved after examining the variation (example: Should the school invest in a computer-based tutoring program for low achieving students in Algebra I after examining the grade distribution? Are more shafts being produced out of specifications that are too big rather than too small?) o Displaying easily the variation in the process (example: Which units are causing the most difficulty for students? Is the variation in a process due to parts that are too long or parts that are too short?) 4

5 Elementary Statistics Dr. Ghamsary Chapter Page Histogram of Test1 Normal Mean 76.8 StDev 1.98 N 50 Frequency Test Probability Plot of Test1 Normal - 95% CI Percent Mean 76.8 StDev 1.98 N 50 AD P-Value Test

6 Elementary Statistics Dr. Ghamsary Chapter Page 6. The frequency polygon o Making decisions about a process, product or procedure that could be improved (example: a frequency polygon for 64 psychology test scores, shown below to the right.) X Frequency Scatterplot of f vs x 1.5 f Midpoints x

7 Elementary Statistics Dr. Ghamsary Chapter Page 7. The Cumulative Frequency Graph (Ogive) Cumulative frequency is used to determine the number of observations that lie above (or below) a particular value. Upper Class Boundaries Cumulative Frequency Scatterplot of Cumulative f vs x 40 Cumulative f Upper Class Boudaries

8 Elementary Statistics Dr. Ghamsary Chapter Page 8 4. The bar chart Bar charts are useful for comparing classes or groups of data. A class or group can have a single category of data or they can be broken down further into multiple categories for greater depth of analysis. Class Grade Frequency F D C B A Frequency F D C Grade B A 8

9 Elementary Statistics Dr. Ghamsary Chapter Page 9 5. Pie Chart o A pie chart is a way of summarizing a set of categorical data or displaying the different values of a given variable (example: percentage distribution). o Pie charts usually show the component parts of a whole. Often you will see a segment of the drawing separated from the rest of the pie in order to emphasize an important piece of information A 8, 16.0% F 6, 1.0% D 9, 18.0% B 15, 30.0% C 1, 4.0% 9

10 Elementary Statistics Dr. Ghamsary Chapter Page Pareto Charts A Pareto chart is used to graphically summarize and display the relative importance of the differences between groups of data Frequency B C D A F 7. Dot plot A dot plot is a visual representation of the similarities between two sequences. Dotplot of Test Test

11 Elementary Statistics Dr. Ghamsary Chapter Page Stem-Leaf o The Stem-and-Leaf Plot summarizes the shape of a set of data (the distribution) and provides extra detail regarding individual values. o They are usually used when there are large amounts of numbers to analyze. Series of scores on sports teams, series of temperatures or rainfall over a period of time, series of classroom test scores are examples of when Stem and Leaf Plots could be used. Stem Leaf Time series Graph Month Price of AOL Jan 65 Feb 60 Mar 58 Apr 6 May 55 Jun 50 Jul 48 Aug 55 Sep 57 Oct 50 Nov 48 Dec 40 Price of MSFT Time Series Plot of AOL, MSFT Variable AOL MSFT 30 Dec Jan Feb Mar Apr May Jun Month Jul Aug Sep Oct Nov 11

12 Elementary Statistics Dr. Ghamsary Chapter Page 1 Type of Distributions: There are several different kinds of distributions, but the following are the most common used in statistics. Symmetric, normal, or bell shape Positively skewed, Right tail, or skewed to the right side. Negatively skewed, Left tail, or skewed to the left side. Uniform Symmetric, Bell Shape, or Normal Distribution

13 Elementary Statistics Dr. Ghamsary Chapter Page 13 Positively skewed Negatively skewed

14 Elementary Statistics Dr. Ghamsary Chapter Page 14 Uniform

15 Elementary Statistics Dr. Ghamsary Chapter Page 15 Test1 Sex Grade Test1 Sex Grade 76 1 C 76 1 C 6 1 D 59 1 F 68 1 D 9 1 A 69 1 D 93 1 A 79 0 C 88 0 B 90 0 A 86 0 B 79 1 C 66 0 D 86 1 B 81 1 B 5 0 F 85 0 B 97 1 A 85 0 B 78 1 C 70 1 C 55 1 F 55 1 F 96 1 A 6 1 D 89 1 B 80 1 B 73 0 C 60 1 D 66 0 D 80 1 B 88 1 B 7 1 C 9 0 A 8 0 B 94 1 A 86 1 B 50 1 F 99 1 A 71 0 C 63 1 D 89 0 B 75 1 C 78 1 C 83 1 B 88 0 B 78 0 C 58 1 F 61 1 D Count Male Sex Female 1=Female 0=Male Grade F D C B A 15 Sex Male Female 1 9 Count F D C B A Grade 70 Sex Male Female Percent F D C B A Grade 15

16 Elementary Statistics Dr. Ghamsary Chapter Page Boxplot of Test1 vs Sex Test Female Sex Male 16

17 Elementary Statistics Dr. Ghamsary Chapter Page 17 Numerical measurements: Statistic:: any value(s) or measure(s) obtained from a sample. Parameter: any value(s) or measure(s) obtained from a specific population. Measures of central tendency: are Mean, Median, and Mode, Mean is defined to be the sum of the scores in the data set divided by the total number of scores. o Sample Mean: is denoted by x, and it is defined by: x n x i = = 1 n i, or simply x x =. n o Population Mean: is denoted by µ, and it is defined by: N xi i µ = = 1, or simply N x µ =. N Note: The sample mean, x is an unbiased estimate of the population mean, µ. Example1: Find the mean of 10, 7, 3, 1, 18. x = = 10. Example: Find the mean of 10, 7, 3, 1, 18, 13, 17, 15, 5, 3 x = = = Example3: Find the mean of scores in the test#1, 004 in data set in this chapter. x = =

18 Elementary Statistics Dr. Ghamsary Chapter Page 18 Median: is defined to be the midpoint of the data set that is arranged from smallest to largest. Example4: Find the median of 10, 7, 3, 1, 15. Solution: First we must sort the data set as follows: 3, 7, 10, 1, 15. The median is 10. Example5: Find the median of 10, 7, 3, 1, 15, 0. Solution: After we sort we get: 3, 7, 10, 1, 15, 0. As we observe, there are middle observations. So to find the median we average these values, namely: Median=(10+1)/ =11. Example6: The median of scores in the test#1, 004 in data is Median = Mode: is defined to be the value in the data set that occurs most frequently. Example7A: Find the mode of 10, 7, 3, 1, 15, 3. Mode is 3. Example7B: Find the mode of 10, 7, 3, 10, 15, 3. Modes are 3 and 10. Example7C: Find the mode of 10, 7, 3, 10, 10, 3. Mode is 10. Example7D: Find the mode of 10, 7, 3, 10, 7, 3. There is no mode, since all values occur with same frequency Example7E: Find the mode of 10, 7, 3, 1, 15, 18. There is no mode, since no values occur more than once. 18

19 Elementary Statistics Dr. Ghamsary Chapter Page 19 Example 8: Find the mean, the median, and the mode of data set: 10, 17, 13, 1, 15, 18, 10, 17, 14, 16, 35, 8,, 17, 3, 1, 15, 8, 10, 0 Solution: First we must sort the data set 10, 10, 10, 1, 1, 13, 14, 15, 15, 16, 17, 17, 17, 18, 0,, 3, 8, 8, o Mean: x = = = o Median: = 16. 5, since there are middle observations o Mode: 10, 17 Example 9: Find the mean, the median, and the mode of data set: 5, 4, 18, 37, 5, 18, 40, 57, 64, 66, 85, 86, 9 85, 88, 9, 67, 33, 75, 85, 48, 60, 80, 60, 50 Example10: Find the mean, the median, and the mode of data set: 1.37, 13.33, 3.67, 1.37,

20 Elementary Statistics Dr. Ghamsary Chapter Page 0 Example11A: Find the mean for the following group data Class Frequency Solution: First we need to find the class marks(midpoints) and then we use the following formula where x : is the midpoint or class mark, and f :is the frequency n:is the number of data points : [ x.f ] x =, n Class Frequency Class marks f x x. f n= f =50 x. f =385 So the mean is x [ x.f ] 385 = = = n 50 0

21 Elementary Statistics Dr. Ghamsary Chapter Page 1 Example11B: Find the mean for the following group data Class Frequency Weighted Average (Mean): The formula in above is also called weighted average or weighted mean. It can also be written as follows: where w is weight and x is the score. x [ w.x] = w Example1: Find the GPA of John who has the following courses with the corresponding units and grades. English Math Spanish 5 units with the grade of A 3 units with the grade of F units with the grade of D Solution: In this problem, x will be the value of the grades and w is the number of units, [ w.x ] [. 4] [. 0] [. 1] w x = = = = = Example13: A teacher is teaching 3 classes: There are 30 students in the first Class with the average of 70 on the final exam. The second class has 40 students with the average of 60 on the final exam. The 3 rd class has 0 students with the average of 80 on the final exam. Find the weighted average of the three classes combined together. Solution: Let x be the average of and w be the number of students. x [ w.x] 70( 30 ) + 60( 40 ) + 80( 0 ) = w = = =

22 Elementary Statistics Dr. Ghamsary Chapter Page Measures of Variation Range Variance Standard Deviation The Range: is defined to be the highest value minus the lowest value in the data set The Variance: is defined by the following: Sample: s variance). Population: variance). = n i= 1 ( x x) i n 1 N i= 1 σ = or ( x µ ) i N s =, or σ = x ( x) n n 1 x d N x i (short cut formula of the sample N (short cut formula of the sample Standard deviation: is the positive square root of the variance. Standard deviation = Variance Sample: s = n ( x ) i x i= 1 n 1, and Population: σ = N ( x ) i µ i= 1 N

23 Elementary Statistics Dr. Ghamsary Chapter Page 3 Example14A: Find the range, variance, and the standard deviation of the following data set. 3, 0, 7, 5, 15. Solution: o Range: Largest- Smallest = 15-0=15 o Variance: If we use the s = n i= 1 ( x x) i n 1, first we need to find the sample mean x. 3 So x = = = 6, then we substitute in the above formula and we get 5 5 s = b3 6g + b0 6g + b7 6g + b5 6g + b15 6g, 5 1 s b 3g b 6g bg 1 b 1g b9g s = s = = 3, So the variance is s = 3. 4 x x x ( ) x x = 5 1, =-3 0-6=-6 7-6=1 5-6= = n ( x x) =0 ( x x) ( x x) =18 i i= s = = = = 3 n o Standard deviation: As we know the standard deviation is positive square root of variance. standard deviation = Variance =

24 Elementary Statistics Dr. Ghamsary Chapter Page 4 But if we use the short cut formula s = x ( x) n n 1 x. x = = 30, first we need to find their sum, x, and their sum of squares, x = = = 308 then we have ( x) ( 30) x s = n = 5 = = = = 3, which is exactly the n same as above Example14B: Find the range, variance, and the standard deviation of the following data set. Solution: 10, 17, 13, 1, 15, 18, 10, 17, 14, 16 8,, 17, 3, 1, 15, 8, 10, 0, 35 4

25 Elementary Statistics Dr. Ghamsary Chapter Page 5 Example15A: Find the standard deviation for the following group data Class Frequency Solution: First will modify the above formula for the variance. But first we need to find the class marks (midpoints) and then we use the following formula s = b i g. x x f n 1 where x : is the midpoint or class mark f : is the frequency n: is the number of data points We already know the mean Class or s = [ x.f ] 3865 x f x = = = n 50 f x x. f x x n 1 ( xf ) n b i g b xi xg. f ( ) = ( ) = ( ) = ( ) = ( ) =34 59 n = f =50 x. f =385 bxi xg. f =

26 Elementary Statistics Dr. Ghamsary Chapter Page 6 After substitution in s = b i g. x x f n 1 we get standard deviation will be s = If we use the short cut formula Class s = x f f x x. f s = =. n 1 ( xf ) n x , and hence the, we need the following table. f (54.5).6 = (64.5)..9 = (74.5).1 = (84.5).15= s n= f =50 x. f ( 385) =385 (94.5).8 =7144 x. f = = 50 = = = = and hence the standard deviation will be s = , which the same as 8 49 the above result. 6

27 Elementary Statistics Dr. Ghamsary Chapter Page 7 Example15B: Find the standard deviation for the following group data Class Frequency

28 Elementary Statistics Dr. Ghamsary Chapter Page 8 Question 1. What will happen to the mean, median, mode, range, and standard deviation if we add a fix number, c, to all values in the data set? Answer. The mean, median, and mode will increase by c units, but the range, and standard deviation will not change. Question. What will happen to the mean, median, mode, range, and standard deviation if we subtract a fix number, c, from all values in the data set? Answer. The mean, median, and mode will decrease by c units, but the range, and standard deviation will not change. Question 3. What will happen to the mean, median, mode, range, and standard deviation if we multiply a fix number, c, to all values in the data set? Answer. The mean, median, and mode will be multiplied by c units, so does to the range, and standard deviation. Example 16: X X+7 X-7 X* = 16+7=3 15+7= 15+7= +7=9 15-7=8 16-7=9 15-7=8 15-7=8-7=15 15*7=105 16*7=11 15*7=105 15*7=105 *7=154 Mean Median Mode Range Sd =3 15+7= 15+7= =9 15-7=8 15-7= *7=11 15*7=105 15*7=105 9*7= *7=4. In general if Y = ax + b, then we have Mean of Y = a. [Mean of X]+b or y = ax + b Standard deviation of Y = a [standard deviation of X], S = a S y X 8

29 Elementary Statistics Dr. Ghamsary Chapter Page 9 Empirical Rule If the distribution of a data is bell shape or normal, then Approximately 68% of scores are one standard deviation away from the mean. They fall in the interval x 1 s, x + 1 s. Approximately 95% of scores are two standard deviation away from the mean. They fall in the interval x s, x + s. Approximately 99.7% of scores are two standard deviation away from the mean. They fall in the interval x 3 s, x + 3 s. Example17. Suppose the IQ scores are normally distributed with the mean of µ = 100 and standard deviation of σ = 15. Then by the empirical rule Approximately 68% of scores are in the interval , to or 85 to 115. Approximately 95% of scores are in the interval 100-(15), to100+(15) or 70 to 130. Approximately 99.7% of scores are in the interval 100-3(15), to100+3(15) or 55 to

30 Elementary Statistics Dr. Ghamsary Chapter Page 30 Coefficient of Variation The coefficient of variation is defined to be the standard deviation divided by the mean. Coefficient of variation (CV) = s. If x is 0 or close to 0, then this measure shall not be used. x Normally this measure is used in the case we have or more groups of data with different units. Example18. Class A Mean =19, and standard deviation= 11 CV=11/19=.085 or 8.5% Class B Mean =150, and standard deviation= 5 CV=5/150=.167 or 16.7% Class C Mean =60, and standard deviation= 15 CV=15/60 =.5 or 5.0% The class C has the greatest relative variation. Measures of Position Standard Scores x x x µ z = or z = s σ, where, x or µ is the mean s or σ is the standard deviation. This value, z, measures the deviation from the mean in number of standard deviation which is also has no unit. Example19. Suppose John is taking 3 classes with the following scores. In which class has he better score? Class A English test score = 145 Mean =19, and standard deviation= 11 Class B Physics test score = 190 Mean =150, and standard deviation= 5 Class C Statistics test score = 88 Mean =60, and standard deviation= 15 Z=(145-19)/11 =1.45 Z=( )/5 = 1.60 Z=(88-60)/15=1.87 So his score in class C is higher relatively. 30

31 Elementary Statistics Dr. Ghamsary Chapter Page 31 Percentiles The percentile corresponding to a given score (X) is denoted by P and it is given by the following formula #of scores less than x P =.100 total number of scores Example0. John has the score of 88 in a class of 0 students. Find the percentile rank of a his score. 81, 65, 75, 76, 78, 6, 63, 65, 70, 90, 61, 75, 76, 79, 58, 88, 8, 95, 90, 67. Solution: In any problem of finding percentile, we must sort the data set from smallest to largest. 58, 61, 6, 63, 65, 65, 67, 70, 75, 75 76, 76, 78, 79, 81, 8, 88, 90, 90, 95. # of scores less than x P = =.100 total number of scores 0 = 80 So john s score has 80 th percentile, which means 80% of all scores are below 88. Finding the Score Corresponding to a Given Percentile Example1. In data set of example 0, find the score corresponding 1 th percentile. Solution: Step1: Make sure data is sorted 58, 61, 6, 63, 65, 65, 67, 70, 75, 75 76, 76, 78, 79, 81, 8, 88, 90, 90, 95 Step: Compute the L = p% of n., where L is the location for the score. In this example L=1%of 0=0.1(0)=.4 or 3. Step3: Go to the data set and pick the score at the 3 rd position which is 6. It is usually written as P 1 =6 31

32 Elementary Statistics Dr. Ghamsary Chapter Page 3 Note: If L is not a whole number, round up to the next whole number. If L is a whole number, use the score as the average of Lth and (L+1)th location score. Example. In data set of example 0, find the score corresponding 40 th percentile. Step1: as before 58, 61, 6, 63, 65, 65, 67, 70, 75, 75 76, 76, 78, 79, 81, 8, 88, 90, 90, 95 Step: L =40% of 0= 0.40(0)=8 which is a whole number so we are going to pick the average of 8 th and 9 th scores. Step3: 8 th score is 70 9 th score is 75 and their average is (70+75)/=7.5. So P 40 =7.5. Deciles: divide the data set into 10 groups. D 1 =10 th percentile which the same as P 10 D =0 th percentile which the same as P 0. D 9 =90 th percentile which the same as P 90 Quartiles: divide the data set into 4 groups. Q 1 =First quartile or 5 th percentile which the same as P 5 Q =second quartile or 50 th percentile which the same as P 50. This is also median Q 3 =third quartile or 75 th percentile which the same as P 75 Inter-Quartile Range (IQR): is the difference between 3 rd and 1 st quartiles and it is denoted by IQR and it is defined by IQR = Q3 Q1. 3

33 Elementary Statistics Dr. Ghamsary Chapter Page 33 Example3. In data set of example 0, find the score corresponding to D Q1 Q3 IQR Outlier: An outlier is an extremely high or an extremely low data value, To check for outlier we compute Q 1-1.5(IQR) and Q (IQR), then if The suspected score is below Q 1-1.5(IQR) or The suspected score is above Q (IQR) Then the score is said to be an outlier. Example4. Is there any outlier in the following data set? Sorted Data

34 Elementary Statistics Dr. Ghamsary Chapter Page 34 Five commonly used Statistics: The five numbers in any data set that is used frequently are Minimum, Q 1, Q, Q 3, Maximum Box plot or box-and-whisker plot: is another graphical representation of any data set. We use the five commonly used statistics to graph the box plot. The box plot can provide answers to the following questions o Is a factor significant? o Does the location differ between subgroups? o Does the variation differ between subgroups? o Are there any outliers? Example5. In data set of example 0, find the 5 common statistics. 1. Minimum: is 58 58, 61, 6, 63, 65, 65, 67, 70, 75, 75 76, 76, 78, 79, 81, 8, 88, 90, 90, 95. Q 1 : L= 5% of 0 =.5(0) = 5. Since this is a whole number we use the average of 5 th and 6 th observation. In above ordered data set we have 5 th score is 65 6 th score is 65 their average is also 65. SO Q 1 = Q : L= 50% of 0 =0.50(0) =10. Again since this is a whole number we use the average of 10 th and 11 th observation. In above ordered data set we have 10 th score is th score is 76 their average is (75+76)/=75.5 SO Q = Q 3 : L= 75% of 0 =0.75(0) =15. This is a whole number we use the average of 15 th and 16 th observation. In above ordered data set we have 15 th score is 81 34

35 Elementary Statistics Dr. Ghamsary Chapter Page th score is 8 their average is (81+8)/=81.5 SO Q 3 = Maximum: is 95. So the five statistics are 58, 65, 75.5, 81.5, and Boxplot of C1 90 C Example6 In data set of example 4, find the 5 common statistics. 35

36 Elementary Statistics Dr. Ghamsary Chapter Page 36 Example7. In data set below use computer to find the descriptive statistics and plot all appropriate charts for all variables that was discussed so far. Test1 Sex Grade Test1 Sex Grade 76 1 C 76 1 C 6 1 D 59 1 F 68 1 D 9 1 A 69 1 D 93 1 A 79 0 C 88 0 B 90 0 A 86 0 B 79 1 C 66 0 D 86 1 B 81 1 B 5 0 F 85 0 B 97 1 A 85 0 B 78 1 C 70 1 C 55 1 F 55 1 F 96 1 A 6 1 D 89 1 B 80 1 B 73 0 C 60 1 D 66 0 D 80 1 B 88 1 B 7 1 C 9 0 A 8 0 B 94 1 A 86 1 B 50 1 F 99 1 A 71 0 C 63 1 D 89 0 B 75 1 C 78 1 C 83 1 B 88 0 B 78 0 C 58 1 F 61 1 D Descriptive Statistics: Test1 Variable Sex N N* Mean SE Mean StDev Minimum Q1 Median Q3 Maximum Test1 Female Male

37 Elementary Statistics Dr. Ghamsary Chapter Page Boxplot of Test1 vs Sex Test Female Sex Male 37