LECTURE 3 - inferential statistics bmaths

Statistics in simplicity

1. LECTURE : statisticS OAF 112: BUSINESS MATHEMATICS AND STATISTICS By; Jafari Selemani [email protected] I 0655 354397 1 By Jafari Selemani - 0655354397

2. What is statistics? a branch of mathematics that provides techniques to analyze whether or not your data is significant (meaningful) Statistical applications are based on probability statements Nothing is “proved” with statistics Statistics are reported Statistics report the probability that similar results would occur if you repeated the experiment

3. Statistics deals with numbers: Need to know nature of numbers collected Continuous variables : type of numbers associated with measuring or weighing; any value in a continuous interval of measurement. Examples: Weight of students, height of plants, time to flowering Discrete variables : type of numbers that are counted or categorical Examples: Numbers of boys, girls, insects, plants

4. Populations and Samples: Population includes all members of a group Example: all 9 th grade students in America Number of 9 th grade students at SR No absolute number Sample Used to make inferences about large populations Samples are a selection of the population Example: 6 th period Accelerated Biology Why the need for statistics? Statistics are used to describe sample populations as estimators of the corresponding population Many times, finding complete information about a population is costly and time consuming. We can use samples to represent a population .

5. Measures of Central Tendency Find the mean Find the median Find the mode Make and interpret a frequency distribution Find the mean of grouped data

6. Data set : a collection of values or measurements that have a common characteristic. Statistic : a standardized, meaningful measure of a set of data that reveals a certain feature or characteristic of the data. Mean : the arithmetic average of a set of data or sum of the values divided by the number of values. Median : the middle value of a data set when the values are arranged in order of size. Mode : the value or values that occur most frequently in a data set . Key Terms

7. 1. Find the Mean A business records its daily sales. These values are an example of a data set . Data sets can be used to: Observe patterns Interpret information Make predictions about future activity From raw (ungrouped) data, Mean is found as; Find the sum of the values. Divide the sum by the total number of values. Mean = sum of values number of values

8. Here’s an example: Sales figures for the last week for the Western Region have been as follows: Monday $4,200 Tuesday $3,980 Wednesday $2,400 Thursday $3,100 Friday $4,600 What is the average daily sales figure? Soln : (4,200 + 3,980 + 2,400 + 3,100 + 4,600) ÷ 5 = 3,656

9. 2. Find the Median Arrange the values in the data set from smallest to largest (or largest to smallest) and select the value in the middle. If the number of values is odd , it will be exactly in the middle. If the number of values is even , identify the two middle values. Add them together and divide by two.

10. Here is an example A recent survey of the used car market for the particular model John was looking for yielded several different prices: $9,400, $11,200, $5,900, $10,000, $4,700, $8,900, $7,800 and $9,200. Required: Find the median price . SOLN: Arrange from highest to lowest : $11,200, $10,000, $9,400, $9,200 , $8,900 , $7,800, $5,900 and $4,700. Calculate the average of the two middle values. (9,200 + 8,900) ÷ 2 = $9,050 or the median price

11. 3. Find the Mode Find the mode in a data set by counting the number of times each value occurs. Identify the value or values that occur most frequently. There may be more than one mode if the same value occurs the same number of times as another value. If no one value appears more than once, there is no mode . EXAMPLE: Results of a placement test in mathematics included the following scores: 65, 80, 90, 85, 95, 85, 80, 70 and 80. Which score occurred the most frequently? 80 is the mode. It appeared three times.

12. Make and Interpret a Frequency Distribution Identify appropriate intervals for the data. Tally the data for the intervals. Count the number in each interval . Key Terms: Class intervals : special categories for grouping the values in a data set. Tally : a mark that is used to count data in class intervals. Class frequency : the number of tallies or values in a class interval. Grouped frequency distribution : a compilation of class intervals, tallies, and class frequencies of a data set.

13. Example: Test scores on the last math test were as follows: 78 84 95 88 99 92 87 94 90 77 REQUIRED: Make a relative frequency distribution using intervals of 75-79, 80-84, 85-89, 90-94, and 95-99.

14. Look at this example 78 84 95 88 99 92 87 94 90 77 Class Class Relative Interval Frequency Calculations Frequency 75-79 2 2/10 20% 80-84 1 1/10 10% 85-89 2 2/10 20% 90-94 3 3/10 30% 95-99 2 2/10 20% Total 10 10/10 100%

15. 4. How to Find the Mean of Grouped Data Make a frequency distribution. Find the products of the midpoint of the interval and the frequency for each interval for all intervals. Find the sum of the frequencies. Find the sum of the products from step 2. Divide the sum of the products by the sum of the frequencies. EXAMPLE: From previous data find the group mean?

16. Look at this example: 78 84 95 88 99 92 87 94 90 77 Product of Class Class Midpoint and Interval Frequency Midpoint Frequency 75-79 2 77 154 80-84 1 82 82 85-89 2 87 174 90-94 3 92 276 95-99 2 97 194 Total 10 880 Mean of the grouped data: 880 ÷ 10 = 88

17. Graphs and Charts: Interpret and draw a bar graph. Interpret and draw a line graph. Interpret and draw a circle graph.

18. 1. Draw and Interpret a Bar Graph Write an appropriate title . Make appropriate labels for bars and scale. The intervals should be equally spaced and include the smallest and largest values. Draw horizontal or vertical bars to represent the data . Bars should be of uniform width. Make additional notes as appropriate to aid interpretation .

19. Here’s an example

20. Interpret and Draw a Line Graph Write an appropriate title . Make and label appropriate horizontal and vertical scales, each with equally spaced intervals . Often , the horizontal scale represents time. Use points to locate data on the graph. Connect data points with line segments or a smooth curve.

21. Here’s an example

22. Interpret and Draw a Circle Graph Write an appropriate title . Find the sum of values in the data set. Represent each value as a fraction or decimal part of the sum of values. For each fraction, find the number of degrees in the sector of the circle to be represented by the fraction or decimal. (100% = 360°) Label each sector of the circle as appropriate.

23. Here’s an example

24. Measures of Dispersion Find the range. Find the standard deviation. Find the variance Key Terms: Measures of central tendency : statistical measurements such as the mean, median or mode that indicate how data groups toward the center. Measures of variation or dispersion : statistical measurement such as the range and standard deviation that indicate how data is dispersed or spread.

25. Range : the difference between the highest and lowest values in a data set. (also called the spread) Deviation from the mean : the difference between a value of a data set and the mean. Standard variation : a statistical measurement that shows how data is spread above and below the mean. Variance : a statistical measurement that is the average of the squared deviations of data from the mean. The square root of the variance is the standard deviation. Square root : the quotient of number which is the product of that number multiplied by itself. The square root of 81 is 9. (9 x 9 = 81) Normal distribution : a characteristic of many data sets that shows that data graphs into a bell-shaped curve around the mean .

26. Quartiles: Data can be divided into four regions that cover the total range of observed values. Cut points for these regions are known as quartiles. In notations, quartiles of a data is the ((n+1)/4) q th observation of the data, where q is the desired quartile and n is the number of observations of data. An example with 15 numbers 3 6 7 11 13 22 30 40 44 50 52 61 68 80 94 Q1 Q2 Q3 The first quartile is Q1=11. The second quartile is Q2=40 (This is also the Median.) The third quartile is Q3=61.

27. Inter-quartile Range: Difference between Q3 and Q1. Inter-quartile range of the previous example is 61- 40=21. The middle half of the ordered data lie between 40 and 61. Coefficient of Variation: The standard deviation of data divided by it’s mean. It is usually expressed in percent. Coefficient of variation =

28. 5. Find the Range in a Data Set Find the highest and lowest values. Find the difference between the two. Example : The grades on the last exam were 78, 99, 87, 84, 60, 77, 80, 88, 92, and 94. The highest value is 99. The lowest value is 60. The difference or the range is 39.

29. Calculation of median – Grouped data: For calculation of median in a continuous frequency distribution the following formula will be employed. Algebraically ,

30. Example: Median of a set Grouped Data in a Distribution of Respondents by age Age Group Frequency of Median class(f) Cumulative frequencies( cf ) 0-20 15 15 20-40 32 47 40-60 54 101 60-80 30 131 80-100 19 150 Total 150

31. Median (M)=40+ 40+ = = 40+0.52X20 = 40+10.37 = 50.37

32. GROUPED MODE : defined it as “the mode of a distribution is the value at the point armed with the item tend to most heavily concentrated. It may be regarded as the most typical of a series of value” The exact value of MODE can be obtained by the following formula. Where; L 1 = Lower class limit of modal class F 1 = Frequency in modal class; F0 = frequency below modal class; F2 = frequency above modal class; i = class interval Z=L 1 +

33. Monthly rent (Rs) Number of Libraries (f) 500-1000 5 1000-1500 10 1500-2000 8 2000-2500 16 2500-3000 14 3000 & Above 12 Total 65 Example: Calculate Mode for the distribution of monthly rent Paid by Libraries in Karnataka

34. Z=2000+ Z =2000+ Z=2400 Z=2000+0.8 ×500=400

35. Find the Standard Deviation The deviation from the mean of a data value is the difference between the value and the mean. Get a clearer picture of the data set by examining how much each data point differs or deviates from the mean . When the value is smaller than the mean, the difference is represented by a negative number indicating it is below or less than the mean . Conversely, if the value is greater than the mean, the difference is represented by a positive number indicating it is above or greater than the mean .

36. Find the mean of a set of data. Mean = Sum of data values Number of values Find the amount that each data value deviates or is different from the mean. Deviation from the mean = Data value - Mean Here’s an example: From the following data set, find the deviation from the mean; Data set: 38, 43, 45, 44

37. Mean = 42.5 1st value: 38 – 42.5 = -4.5 below the mean 2nd value: 43 – 42.5 = 0.5 above the mean 3rd value: 45 – 42.5 = 2.5 above the mean 4th value: 44 – 42.5 = 1.5 above the mean Interpret the information: One value is below the mean and its deviation is -4.5. Three values are above the mean and the sum of those deviations is 4.5. The sum of all deviations from the mean is zero. This is true of all data sets . We have not gained any statistical insight or new information by analyzing the sum of the deviations from the mean.

38. Find the standard deviation of a set of data A statistical measure called the standard deviation uses the square of each deviation from the mean. The square of a negative value is always positive. The squared deviations are averaged (mean) and the result is called the variance. The square root is taken of the variance so that the result can be interpreted within the context of the problem. This formula averages the values by dividing by one less than the number of values (n-1). Several calculations are necessary and are best organized in a table .

39. Steps in finding SD: Find the mean. Find the deviation of each value from the mean. Square each deviation. Find the sum of the squared deviations. Divide the sum of the squared deviations by one less than the number of values in the data set. This amount is called the variance. Find the standard deviation by taking the square root of the variance.

40. EXAMPLE: Find the standard deviation for the following data set: 18 22 29 27 Deviation Squares of Value Mean from Mean Deviation 18 24 18 – 24 = -6 -6 x -6 = 36 22 24 22 – 24 = -2 -2 x -2 = 4 29 24 29 – 24 = 5 5 x 5 = 25 27 24 27 – 24 = 3 3 x 3 = 9 Sum of Squared Deviations 74

41. Variance = sum of squared deviations n – 1 Variance = 74 ÷ 3 = 24.666667 Standard deviation = square root of the variance Standard deviation = 4.97 rounded

42. A large variance means that the individual scores (data) of the sample deviate a lot from the mean. A small variance indicates the scores (data) deviate little from the mean Variance helps to characterize the data concerning a sample by indicating the degree to which individual members within the sample vary from the mean.

43. Probability Distributions: Inferential statistical methods use sample data to make predictions about the values of useful summary descriptions, called parameters, of the population of interest. This part treats parameters as known numbers. We first define the term probability , using a relative frequency approach . The probability distribution of the random variable X lists the possible outcomes together with their probabilities the variable X can have . From the probability distribution, The mean and the standard deviation of the discrete random variable are defined in the following ways.

44. The variance and standard deviation will be;