Advance Statistics Learning Topics -Jobelle M Quilana Section 7001-1-1-B.pptx

Lecture in advance statistics

1. Master of Arts in Education - Major in Educational Management JOBELLE M. QUILANA Section 7001-1-1-B ADVANCED STATISTICS

2. LEARNING TOPICS: Summation Notation Measures of Central Tendency Arithmetic Mean Median

3. Summation Notation  

4. Summation notation (or sigma notation) allows us to write a long sum in a single expression. This is the sigma symbol: . It tells us that we are summing up something.  

5. Summation: A series of addition read as “the summation of sub , is from 1 to .  

6. Summation This means taking the sum of number of observations or values of the variable represented by . The subscript represents the order of an observation, whether it is the first, second, third or the last. The notation under the summation sign , denotes the lower limit and indicates the start of counting. The number above, , is the upper limit and tells the total number of observations to be added.  

7. Summation Thus, the symbol indicates the sum of the first three variables, while the symbol indicates the sum of the second to fifth values of .  

8. Suppose the grades obtained by five high school students in a high school mathematics test are as follows: and . There are five observations, hence . These are five values of represented as and . The symbol used to represent the sum of these five numbers would then be .  

9. Summation Rules Summation of constants  

10. Summation Rules Summation of a Sum =      

11. Summations Rules Summation of a Variable and a Constant     = =  

12. Summation Rules Sum of a Product   +   +  

13. Summation Rules Summation of the Product of a Constant and a Variable          

14. Summation Rule Square of the Sum of Variables      

15. Summation Rule Sum of the Squares of Variables     + +  

16. MEASURES OF CENTRAL TENDENCY

17. MEASURES OF CENTRAL TENDENCY A measure of central tendency is a summary measure that attempts to describe a whole set of data with a single value that represents the middle or center of its distribution. There are three main measures of central tendency: mean median mode

18. The mean is the sum of the value of each observation in a dataset divided by the number of observations. This is also known as the arithmetic average. Example: Looking at the retirement age distribution again: 54, 54, 54, 55, 56, 57, 57, 58, 58, 60, 60 The mean is calculated by adding together all the values (54+54+54+55+56+57+57+58+58+60+60 = 623) and dividing by the number of observations (11) which equals 56.6 years. Mean

19. Median The median is the middle value in distribution when the values are arranged in ascending or descending order. The median divides the distribution in half (there are 50% of observations on either side of the median value). In a distribution with an odd number of observations, the median value is the middle value. When the distribution has an even number of observations, the median value is the mean of the two middle values.

20. Median Example: Looking at the retirement age distribution below (which has 11 observations), the median is the middle value, which is 57 years. 54, 54, 54, 55, 56, 57, 57, 58, 58, 60, 60 In the distribution below, the two middle values are 56 and 57, therefore the median equals 56.5 years. 52, 54, 54, 54, 55, 56, 57, 57, 58, 58, 60, 60

21. Mode The mode is the most commonly occurring value in a distribution. Example: Consider this dataset showing the retirement age of 11 people, in whole years: 54, 54, 54, 55, 56, 57, 57, 58, 58, 60, 60 This table shows a simple frequency distribution of the retirement age data. The most commonly occurring value is 54, therefore the mode of this distribution is 54 years.

22. Weighted mean Used when some data values are more important than the others. Formula: Weighted Mean = Where: x = data/each number in the data w = weight/importance

23. Weighted mean Example: Find Carl’s GPA using weighted mean Subject Grade (x) Units (w) English 4 3 Math 4 3 Physics 3 4 Statistics 2.33 3 Chemistry 2.67 2 Solution: Weighted Mean = = (3x4) + (3x4) + (4x3) + (3x2.33) + (2x2.67)/3+3+4+3+2 = 48.33/15 Weighted Mean = 3.22

24. Midrange The value that is halfway between the minimum data value and the maximum data value. Formula: Midrange = minimum value + maximum value 2 Example: Find the midrange of the following daily temperatures which were recorded at 3 hour interval. 52˚, 65˚, 71˚, 74˚, 76˚, 75˚, 68˚, 57˚, 54˚ Solution: first, get the minimum and maximum value from the given data. (min val = 52˚, max val = 76˚) Midrange = 52˚+76˚ = 128/2 = 64˚ 2

25. Arithmetic Mean

26. Arithmetic Mean The arithmetic mean in statistics, is nothing but the ratio of all observations to the total number of observations in a data set. Some of the examples include the average rainfall of a place, the average income of employees in an organization. We often come across statements like "the average monthly income of a family is P15,000 or the average monthly rainfall of a place is 1000 mm" quite often. Average is typically referred to as arithmetic mean.

27. Arithmetic Mean Arithmetic mean is often referred to as the mean or arithmetic average. It is calculated by adding all the numbers in a given data set and then dividing it by the total number of items within that set. The formula for calculating arithmetic mean is (sum of all observations)/(number of observations). For example, the arithmetic mean of a set of numbers {10, 20, 30, 40} is (10 + 20 + 30 + 40)/4 = 25.

28. The arithmetic mean of ungrouped data is calculated using the formula: Mean x̄ = Sum of all observations / Number of observations Example: Compute the arithmetic mean of the first 6 odd natural numbers. Solution: The first 6 odd natural numbers: 1, 3, 5, 7, 9, 11 x̄ = (1+3+5+7+9+11) / 6 = 36/6 = 6. Calculating Arithmetic Mean for Ungrouped Data

29. Calculating Arithmetic Mean for Grouped Data There are three methods to calculate the arithmetic mean for grouped data. Direct method Short-cut method Step-deviation method

30. Direct Method Let x₁, x₂, x₃ ……xₙ be the observations with the frequency f₁, f₂, f₃ ……fₙ. Then, arithmetic mean is calculated using the formula: x̄ = ( x₁f ₁+ x₂f ₂+......+ xₙf ₙ) / ∑fi Here, f₁+ f₂ + ....fₙ = ∑fi indicates the sum of all frequencies.

31. Direct Method Example I (discrete grouped data): Find the arithmetic mean of the following distribution: x 10 30 50 70 89 f 7 8 10 15 10 x i f i x i f i 10 7 10×7 = 70 30 8 30×8 = 240 50 10 50×10 = 500 70 15 70×15 = 1050 89 10 89×10 = 890 Total ∑f i =50 ∑ x i f i =2750 Add up all the ( x i f i ) values to obtain ∑ x i f i . Add up all the f i values to get ∑f i Now, use the arithmetic mean formula. x̄ = ∑ x i f i / ∑f i = 2750/50 = 55 Arithmetic mean = 55.

32. Direct Method Example II (continuous class intervals): Let's try finding the mean of the following distribution: Class-Interval 15-25 25-35 35-45 45-55 55-65 65-75 75-85 Frequency 6 11 7 4 4 2 1 Solution: When the data is presented in the form of class intervals, the mid-point of each class (also called class mark) is considered for calculating the arithmetic mean. Note: Class Mark = (Upper limit + Lower limit) / 2 Class- Interval Class Mark (x i ) Frequency (f i ) x i f i 15-25 20 6 120 25-35 30 11 330 35-45 40 7 280 45-55 50 4 200 55-65 60 4 240 65-75 70 2 140 75-85 80 1 80   Total 35 1390 x̄ = ∑ xifi / ∑fi = 1390/35 = 39.71. Arithmetic mean = 39.71

33. Short-cut Method The short-cut method is called as assumed mean method or change of origin method. The following steps describe this method. Step1: Calculate the class marks (mid-point) of each class (x i ). Step2: Let A denote the assumed mean of the data. Step3: Find deviation (d i ) = x i – A Step4: Use the formula: x̄ = A + (∑ f i d i /∑f i )

34. Example:  Calculate the mean of the following using the short-cut method. Short-cut Method Class-Intervals 45-50 50-55 55-60 60-65 65-70 70-75 75-80 Frequency 5 8 30 25 14 12 6 Solution:  Let us make the calculation table. Let the assumed mean be  A  = 62.5 Note: A is chosen from the x i values. Usually, the value which is around the middle is taken. Class- Interval Classmark/ Mid-points (x i ) f i d i  = (x i  - A) f i d i 45-50 47.5 5 47.5-62.5 =-15 -75 50-55 52.5 8 52.5-62.5 =-10 -80 55-60 57.5 30 57.5-62.5 =-5 -150 60-65 62.5 25 62.5-62.5 =0 65-70 67.5 14 67.5-62.5 =5 70 70-75 72.5 12 72.5-62.5 =10 120 75-80 77.5 6 77.5-62.5 =15 90     ∑f i =100   ∑ f i d i = -25 Now we use the formula, x̄ = A + (∑ fidi /∑fi) = 62.5 + (−25/100) = 62.5 − 0.25 = 62.25

35. Step Deviation Method This is also called the change of origin or scale method. The following steps describe this method: Step 1: Calculate the class marks of each class (x i ). Step 2: Let A denote the assumed mean of the data. Step 3: Find u i = (x i −A)/h, where h is the class size. Step 4: Use the formula to find the arithmetic mean: x̄ = A + h × (∑ f i u i /∑f i )

36. Step Deviation Method Class Intervals 0-10 10-20 20-30 30-40 40-50 50-60 60-70 Total Frequency 4 4 7 10 12 8 5 50 Example:  Consider the following example to understand this method. Find the arithmetic mean of the following using the step-deviation method. Solution:  To find the mean, we first have to find the class marks and decide A (assumed mean). Let A = 35 Here h (class width) = 10 C.I. x i f i u i = x i −Ah​​ x i −Ah f i u i 0-10 5 4 -3 4 x (-3)=-12 10-20 15 4 -2 4 x (-2)=-8 20-30 25 7 -1 7 x (-1)=-7 30-40 35 10 10 x 0= 0 40-50 45 12 1 12 x 1=12 50-60 55 8 2 8 x 2=16 60-70 65 5 3 5 x 3=15 Total   ∑fi=50   ∑ fiui =16 Using arithmetic mean formula: x̄ = A + h × (∑ f i u i /∑f i ) =35 + (16/50) ×10 = 35 + 3.2 = 38.2

37. Median

38. Median Median, in statistics, is the middle value of the given list of data when arranged in an order. The arrangement of data or observations can be made either in ascending order or descending order. The median of a set of data is the middlemost number or center value in the set. The median is also the number that is halfway into the set.

39. Median formula Odd Number of Observations If the total number of observations (n) given is odd, then the formula to calculate the median is: Even Number of Observations If the total number of observations (n) is even, then the median formula is:

40. Median formula Example 1: Imagine that a top running athlete in a typical 200-metre training session runs in the following times: 26.1 seconds, 25.6 seconds, 25.7 seconds, 25.2 seconds, 25.0 seconds, 27.8 seconds and 24.1 seconds. How would you calculate his median time? Solution: Let’s start with arranging the values in increasing order: Rank Times (in seconds) 1 24.1 2 25.0 3 25.2 4 25.6 5 25.7 6 26.1 7 27.8 There are n = 7 data points, which is an uneven number. The median will be the value of the data points of rank (n + 1) ÷ 2 = (7 + 1) ÷ 2 = 4. The median time is 25.6 seconds.

41. Median formula If the number of data points is even, the median will be the average of the data point of rank n ÷ 2 and the data point of rank (n ÷ 2) + 1. Example 2: Now suppose that the athlete runs his eighth 200-metre run with a time of 24.7 seconds. What is his median time now? Solution: Let’s start with arranging the values in increasing order: Rank Times (in seconds) 1 24.1 2 24.7 3 25.0 4 25.2 5 25.6 6 25.7 7 26.1 8 27.8 There are now n = 8 data points, an even number. The median is the mean between the data point of rank n ÷ 2 = 8 ÷ 2 = 4 and the data point of rank (n ÷ 2) + 1 = (8 ÷ 2) +1 = 5 Therefore, the median time is (25.2 + 25.6) ÷ 2 = 25.4 seconds.

42. References https://www.khanacademy.org/math/ap-calculus-ab/ab-integration-new/ab-6-3/a/review-summation-notation https://www.scribd.com/presentation/340528254/21-4-Summation-Notation-%CE%A3 https://www.abs.gov.au/statistics/understanding-statistics/statistical-terms-and-concepts/measures-central-tendency https://www.youtube.com/watch?v=dlwvRDibAd8 https://www.cuemath.com/data/arithmetic-mean/ https://byjus.com/maths/median/ https://www150.statcan.gc.ca/n1/edu/power-pouvoir/ch11/median-mediane/5214872-eng.htm