Statistics. a branch of mathematics dealing with the collection, analysis, interpretation, and presentation of masses of numerical data

Transcription

1 Statistics a branch of mathematics dealing with the collection, analysis, interpretation, and presentation of masses of numerical data

2 Lesson #1: Introduction to Statistics Measures of Central Tendency: mean, median and mode MEAN: determined by dividing the sum of all the values in a data set by the number of values in the set For example: Data set A) 2, 3, 4, 5, 6 (the mean is 20/5=4).

3 Lesson #1: Introduction to Statistics Measures of Central Tendency: mean, median and mode MEDIAN: represented by the middle value of an ordered data set (when the middle value is two values, the mean of those values is used). For example: Data set A) 3, 3, 4, 5, 6 (the median is 4). Data set B) 3, 3, 5, 6 (the median is also 4).

4 Lesson #1: Introduction to Statistics Measures of Central Tendency: mean, median and mode MODE: represented by the value that occurs most often in a data set (when there is more than one value that occurs most often, there is no mode). For example: Data set A) 2, 2, 4, 5, 6 (the mode is 2). Data set B) 2, 2, 3, 4,4, 5 (there is no mode).

5 Lesson #1: Introduction to Statistics RANGE: the difference between the maximum value and the minimum value in a data set OUTLIERS: a value in a data set that is very different from other values in the set

6 For Example: The following is a set of 31 scores achieved by students on an examination: Calculate the mean, median, mode and range for the data. Are there any outliers?

7 Tutorial #1 Calculator: Using 1 Variables Stats Step 1. Input Data from question. Enter Stat-Edit Input data into L1 Step 2. Solve for mean and median, mode and range. Enter Stat-Calc-1-Variable Stats-Enter, Enter NOTE: Stat - SortA( - 2 nd 1 enter (this can help with mode)

8 Example. Using this data: 5, 32, 34, 34, 34, 36, 36, 36, 36, 38, 38, 40, 42 find the Mean, Median, and Mode and Range. x x min x 5 max x 42 Mean Sum Square of the sum sx x n 13 Minimum Q1 34 med 36 Q3 38 Maximum Sample Standard Deviation Population Standard Deviation(use this one) Number of data items First Quartile Median Third Quartile Remember (sigma) means to find the sum.

9 Example. Using this data: 5, 32, 34, 34, 34, 36, 36, 36, 36, 38, 38, 40, 42 find the Mean, Median, and Mode and Range. Mean Median x 33.9 med 36 Mode find the most frequent number 36 Range = 42 5 = 37 Minimum min x 5 Maximum max x 42

10 Data Set: 5, 32, 34, 34, 34, 36, 36, 36, 36, 38, 38, 40, 42 Which measure of central tendency (mean, median or mode) best represents the data set?

11 Central tendency Central tendency refers to the idea that there is one number that best summarizes the entire set of measurements, a number that is in some way "central" to the set.

12 Example: The GPA s for 15 students. They range from 0 (failing) to 4 (an A). John Mary George Beth Sam Judy Fritz Kate Dave Jenny Mike Sue Don Ellen Orville Which measure of central tendency (mean, median or mode) best represents the data set?

13 Which measure of central tendency (mean, median or mode) best represents the data set? The mode. The mode is the measurement that has the greatest frequency, the one you found the most of. Although it isn't used that much, it is useful when differences are rare or when the differences are non numerical. The mode for our example is 2.8. It is the grade with the most people (3).

14 The median. The median is the number at which half your measurements are more than that number and half are less than that number. The median is actually a better measure of centrality than the mean if your data are skewed, meaning lopsided. If, for example, you have a dozen ordinary folks and one millionaire, the distribution of their wealth would be lopsided towards the ordinary people, and the millionaire would be an outlier, or highly deviant member of the group. The millionaire would influence the mean a great deal, making it seem like all the members of the group are doing quite well. The median would actually be closer to the mean of all the people other than the millionaire. The median for our example is 2.8. Half the people scored lower, and half higher (and one exactly).

15 The mean. The mean is just the average. It is the sum of all your measurements, divided by the number of measurements. This is the most used measure of central tendency, because of its mathematical qualities. It works best if the data is distributed very evenly across the range, or is distributed in the form of a normal or bell-shaped curve. The mean or average for our example: 2.92.

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17 Statistical Reasoning Central Tendency Assignment: Workbook p #1-8 Due: At the end of the class *Please show work on loose leaf paper. Keep work neat and provide answers in sentence statements.

18 Summary of when to use the mean, median and mode Please use the following summary table to know what the best measure of central tendency is with respect to the different types of variable. Type of Variable Best measure of central tendency Nominal Mode Ordinal Median Interval/Ratio (not skewed) Mean Interval/Ratio (skewed) Median

19 Nominal Measurement the numerical values just name the attribute uniquely. No ordering of the cases is implied. For example: Jersey numbers in basketball A player with number 30 is not more of anything than a player with number 15, and is certainly not twice whatever number 15 is. Ordinal Measurement the attributes can be rank-ordered. Distances between attributes do not have any meaning. For example: A survey on Educational Attainment (0=less than H.S.; 1=some H.S.; 2=H.S. degree; 3=some college; 4=college degree; 5=post college) Higher numbers mean more education. But distance from 0 to 1 is not same as 3 to 4. The interval between values is not interpretable in an ordinal measure.

20 Interval Measurement the distance between attributes does have meaning (these distributions may be normal or skewed) For example: Temperature (in Celsius) The distance from is same as distance from The interval between values is interpretable. Because of this, it makes sense to compute an average of an interval variable, where it doesn t make sense to do so for ordinal scales.

21 Eastman Region worst in province for deervehicle collisions: Manitoba Public Insurance releases Top 5 list The Eastman Region which includes communities Steinbach, Lac du Bonnet, Pine Falls, Birds Hill Park area and St. Malo holds the dubious distinction of being the province's No. 1 area for vehicle-deer collisions. There are about 1,800 collisions yearly within this region, based on statistics collected by Manitoba Public Insurance from 2007 to 2011.

22 Motorists are encouraged to exercise extra care and caution as they head into November - the worst month of the year for deer-vehicle collisions. Rounding out the province's Top 5 deer/vehicle collision regions: Eastman Region 1,800 yearly collisions Brandon/Westman 1,500 yearly collisions Interlake 1,000 yearly collisions Pembina Valley 600 yearly collisions Central Plains and Parkland 500 yearly collisions each On average, there are about 6,800 vehicle-deer collisions each year in Manitoba, according to the Manitoba Public Insurance claims data.

23 In efforts to educate Manitoba motorists about the high-risk areas, two maps, showing the highest-risk and high-risk areas rural Manitoba and Winnipeg are available on the Manitoba Public Insurance website: Winnipeg Area Map nnipegbands.pdf Manitoba Rural Area Map anitobazones.pdf MPI Site

24 Lesson #2: Frequency Tables, Histograms and Frequency Polygons Frequency Tables The frequency is the number of times each data occurs. Simply record tally marks for how often a particular score occurs. That is the frequency. Note: When creating tally marks, every fifth tally should be a stroke through the four before making it easier to total afterwards.

25 Lesson #2: Frequency Tables, Histograms and Frequency Polygons Frequency Tables The number of calls from motorists per day for roadside service was recorded for the month of December The results were as follows: Set up a frequency table for this set of data values and determine the measures of central tendency.

26 Lesson #2: Frequency Tables Note: When selecting interval range/size, use the following rule of thumb: range 10 = interval Reminder: Range = max-min value Frequency Tables, Histograms and Frequency Polygons Road side service calls Tally Frequency

27 Tutorial #2 Calculator: Using 1 Variables Stats with a Frequency Table Steps: 1. Input Frequency Table from question. Stat-Edit Input data into L1, L2 (L1 scores*, L2 frequency) *Note: if entering scores as intervals, use the median value of each interval 2. Solve for mean and median, mode and range. Stat-Calc-1-Variable Stats (L1, L2) Enter, Done

28 Lesson #2: Frequency Tables, Histograms and Frequency Polygons Histograms and Frequency Polygons In addition to finding statistical measures such as the mean and mode, it sometimes helps to see a picture of the data. This can be done with a Histogram or a Frequency Polygon. But first the data needs to be organized. This can be done with a Frequency Table. From a Frequency Table, you can create a Histogram or a Frequency Polygon.

29 Lesson #2: Histograms Plot frequency data on the vertical axis Intervals on the horizontal axis In other words: y-axis is the frequency x-axis is the data (in intervals) Intervals are the range of values you allow for each bar of your graph (i.e. pictured here test scores with an interval of 10 percentage points) Frequency Tables, Histograms and Frequency Polygons

30 Lesson #2: Frequency Tables, Histograms, and Frequency Polygons Example: These are grades from Professor Eyesore s Stats Class: Create a frequency table in order to create a histogram to represent the class grades.

31 Lesson #2: Frequency Tables, Histograms, and Frequency Polygons Grades - % Tally Frequency

32 Lesson #2: Frequency Tables, Histograms, and Frequency Polygons Grades - % Tally Frequency

33 Tutorial #3 Calculator: Creating a Histogram Steps: 1. After inputting data into lists (L1 scores, L2 frequency), go to Stat Plots and set up graph 2 nd Y=, 1: Plot1 choose On, histogram, Xlist: L1, and Freq: L2 2. Adjust the Window Settings to view the histogram WINDOW Xmin/Xmax (min/max scores ) Ymin/Ymax (min/max frequency) Xscl (interval range) Yscl=1 (frequency) *Press the TRACE key and use your arrow keys to view frequencies

34 WINDOW Xmin: Xmax: Ymin: Ymax: Xscl: Yscl: 0 (min score) 120 (max score) 0 (min frequency) 15 (max frequency) 20 (interval range) 5 (frequency) *note: keep at 1

35 WINDOW Xmin: Xmax: Ymin: Ymax: Xscl: Yscl: 0 (min score) 60 (max score) 0 (min frequency) 14 (max frequency) 5 (interval range) 2 (frequency) *note: keep at 1

36 Tutorial #4 Calculator: Creating a Frequency Polygon Steps: 1. After inputting data into lists (L1 scores, L2 frequency), go to Stat Plots and set up graph 2 nd Y=, 1: Plot1 choose On, frequency polygon, Xlist: L1, Freq: L2, Mark: 2. Adjust the Window Settings to view the frequency polygon WINDOW Xmin/Xmax (min/max scores ) Ymin/Ymax (min/max frequency) Xscl (interval range) Yscl=1 (frequency)

37 Statistical Reasoning Frequency Tables and Histogram Assignment: Workbook p #1-3 (use your graphing calculator to construct histograms, not the grid provided) Due: At the end of the class *Please show work on loose leaf paper. Keep work neat and provide answers in sentence statements.

38 A Review on How to Create a Histogram on the TI-83/84 Graphing Calculator age&v=by0qu-yybja Note: If you have a list of values (no frequency table) you can enter all data into List 1 (just like the video) ~ OR ~ If you have a list of data and a frequency list (a frequency table) you must enter your data into List 1 and your frequencies into List 2

39 Requirements: Labeling a Histogram 1. Title the x-axis Age 2. Title the y-axis Frequency or People 3. Label the x-axis intervals Note: the intervals are 10 years apart 4. Label the y-axis increments Note: the increments are by Label the histogram with frequency values Note: the numbers above each bar

40 Requirements: 1. Title the x-axis Class Marks 2. Title the y-axis Frequency or People 3. Label the x-axis intervals Note: the intervals are 20 marks apart 4. Label the y-axis increments Note: the increments are by 5 5. Label the polygon with frequency values Note: the numbers above each dot Labeling a Frequency Polygon

41 Ensure you have your test notes. You may now begin the quiz. Good luck!

42 Standard Deviation Measures of Dispersion: Dispersion: a measure that varies by the spread among the data in a set For example: If all the data in a set is identical, the dispersion has a value of 0. (Note: It increases in value as the data becomes more spread out)

43 Standard Deviation Measures of Dispersion: Deviation: the difference between a data value and the mean for the same set of data For example: The difference between a student s test score and the class average (mean).

44 Standard Deviation Measures of Dispersion: Standard Deviation (σx): a measure of the dispersion or scatter of data values in relation to the mean For example: A low SD indicates that most data values are close to the mean. A high SD indicates that most data values are scattered farther from the mean.

45 Standard Deviation, σ x, is a measure of the variability of data. It tells how widely spread the data is. The higher the standard deviation, the more varied or spread the numbers. For example: You would expect the marks in a grade 12 Applied class to be less varied than those in a grade 9 Math class because of streaming/course selection. For example: A manufacturer of light bulbs would not want the life of their bulbs to vary greatly, so they would aim for a lower standard deviation.

46 How to Calculate Standard Deviation using a formula ailpage&v=qqoyy_njflu

47 The formula for standard deviation is: x x x 2 n

48 The steps to calculate standard deviation are as follows: 1. Find the mean(average) of the numbers. 2. Find the difference between each number and the mean. 3. Find the squares of the differences. 4. Find the sum of these squares. 5. Divide the sum of the squares by n. 6. Find the square root of this number.

49 The steps to calculate standard deviation are as follows: 1. Find the mean(average) of the numbers. 2. Find the difference between each number and the mean.

50 The steps to calculate standard deviation are as follows: 3. Find the squares of the differences. 4. Find the sum of these squares.

51 The steps to calculate standard deviation are as follows: 5. Divide the sum of the squares by n. 6. Find the square root of this number.

52 Example: Ten hockey fans were surveyed to determine the number of hockey games they attended throughout the season. The results were: 5, 6, 7, 8, 8, 9, 9, 9, 10, 11. Find the standard deviation of this set of data.

53 Solution: Complete the table for steps 1, 2, 3, 4, 5, and Sum Total: Number of Games Difference from Mean Square of Difference (x) x x 2 Sum Total: x x Pieces of Data: n - 1 Mean: Sum of Squares n - 1 Square Root:

54 For Example: The prices of 10 cars available for sale at two different dealerships are listed below. Dealership A: $22 500, $31 400, $20 000, $33 550, $18 700, $20 500, $15 500, $42 200, $12 300, $ Dealership B: $45 500, $42 000, $43 500, $39 900, $38 900, $62 500, $55 000, $48 550, $50 000, $ a) Calculate the mean and standard deviation for each set of data. b) Which dealership s prices have the greatest standard deviation?

55 For each of the following frequency tables, calculate the mean and determine which has the least and greatest standard deviation. NOTE: Input the x values into L1 and the y values into L2. Go to Stat-Calc-1 Var Stats L1, L2.

56 Two companies, A and B, manufacture ball bearings. A sample of 100 ball bearings is taken from each company and the diameters measured accurately to the nearest 0.5 mm. The results are shown in the tables below. a) Calculate the mean and standard deviation for each set of data.

57 Two companies, A and B, manufacture ball bearings. A sample of 100 ball bearings is taken from each company and the diameters measured accurately to the nearest 0.5 mm. The results are shown in the tables below. b) Which company manufactures ball bearings with the most uniform size?

58 Two companies, A and B, manufacture ball bearings. A sample of 100 ball bearings is taken from each company and the diameters measured accurately to the nearest 0.5 mm. The results are shown in the tables below. c) If the machines are to be set to produce ball bearings with a diameter of 6.5 mm, which company s machine would be the easiest to adjust? Explain.

59 Statistical Reasoning Standard Deviation Assignment: Workbook p #1-5 Due: At the end of the class *Please show work on loose leaf paper. Keep work neat and provide answers in sentence statements. OVERDUE ASSIGNMENTS: p #1-8 p #1-3

60 Normal Distribution Properties of Normal Distribution: Forms a symmetric bell-shaped curve 50% of the scores lie above and 50% below the midpoint of the distribution Mean, median, and mode are located at the middle of the distribution curve

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62 Normal Distribution Rule: 0.15% 2.35% 2.35% 0.15% 99.7%

63 (to the nearest mm) Enter the data in list L1 of your calculator and calculate the mean and standard deviation. Round your answers to the nearest whole value.

64 99.7%

65 Create a normal distribution curve using the mean and standard deviation. Values below 0mm are not reasonable x = 42 σ = 24

66 If the distribution is normal, find the proportion of months with precipitation that is: Less than 66 mm. Left of +1 SD Left of the mean is 50% Between mean and +1SD is 34% Therefore, less than 66 mm occurs 84% of the time Greater than 18 mm. Right of -1 SD Right of the mean is 50% Between mean and -1SD is 34% Therefore, greater than 18 mm occurs 84% of the time Between 42 mm and 90 mm. Between the mean and +1SD is 34% Between the +1SD and +2SD is 13.5% Therefore, between 42 mm and 90 mm occurs 47.5% of the time

67 Create a normal distribution curve using the mean and standard deviation. 50% 34% Values below 0mm are not reasonable

68 Create a normal distribution curve using the mean and standard deviation. Values below 0mm are not reasonable 34% 50%

69 Statistical Reasoning Normal Distribution Assignment: Workbook p #1-5 Due: At the end of the class *Please show work on loose leaf paper. Keep work neat and provide answers in sentence statements. OVERDUE ASSIGNMENTS: p #1-8 p #1-3 p #1-5

70 Standard Normal Distribution Properties of Standard Normal Distribution: Mean of 0 and Standard Deviation of 1

71 Standard Normal Distribution Z-Scores A z-score is a standardized value that indicates the number of standard deviations a data is above or below the mean Positive z-score indicates a data point above the mean. Negative z-scores indicates a data point below the mean.

72 You can use the following methods to determine Z-Scores: Using Z-Score Table (p ) Using the z-score formula Using your calculator ON YOUR CALCULATOR: To calculate the %: 2 nd VARS (DISTR) 2: normalcdf(lower z-score, upper z-score)

73 ON YOUR CALCULATOR: To calculate the z-score: 2 nd VARS (DISTR) 3: invnorm(% as a decimal) *The % must be entered as the percentage of the area to the LEFT of the z-score. Area to the LEFT = 16% Therefore, use 16% (or 0.16) when entering into the calculator.

74 ON YOUR CALCULATOR: To calculate the z-score: 2 nd VARS (DISTR) 3: invnorm(% as a decimal) *For % of the area to the RIGHT you must calculate 100 subtract the area to the LEFT Area to the RIGHT = 5% Area to the LEFT = 100% 5% Area to the LEFT = 95% Therefore, use 95% (or 0.95) when entering into the calculator.

75 Example: Determine the area under the curve (percentages) for each of the following intervals. Note: Because these are standard values, you can use the percentages of the normal distribution to answer (no calculation required).

76 Determine the percentage of the data: a) Below a z-score of 1.3 b) Above a z-score of 2.5 c) Between a z-score of -1.5 and 0.7 Note: Because these are NOT standard values, you can use your calculator (or the z-score tables) to answer the question.

77 Standard Normal Distribution Z-Scores Z-scores can be used to compare different sets of data that do not have the same mean or standard deviation Z-Score Formula: x μ z = σ Note: μ = x

78 Ex. Calculate the z-score for each of the following. µ = 80, σ = 5, and x = z = 5 µ= 45.5, σ = 2.5, and x = 50 z = 4 z = z = 1.8

79 If two candidates for a scholarship have taken different tests that are equally reputable but have different means and standard deviations, their results may be compared by considering the z-scores. Luke has scored 206 in a test where the mean score is 190 and the standard deviation is 8. Bill has scored 91 in a test that has a mean of 81 and a standard deviation of 4. Who has the better score?

80 Luke has scored 206 in a test where the mean score is 190 and the standard deviation is 8. z = z = 2 Bill has scored 91 in a test that has a mean of 81 and a standard deviation of 4. z = z = 2.5

81 Luke has a z-score of 2. Bill has a z-score of 2.5. Who has the better score?

82 Statistical Reasoning Z-Scores Assignment: Workbook p #1-6 Due: At the end of the class *Please show work on loose leaf paper. Keep work neat and provide answers in sentence statements. OVERDUE ASSIGNMENTS: p #1-5 p #1-8 p #1-3 p #1-5

83 Margin of Error The possible difference between the estimate of the value you re trying to determine, as determined from a random sample, and the true value for the population The margin of error is generally expressed as a plus or minus value, such as ±5%

84 Confidence Interval The interval in which the true value you re trying to determine is estimated to lie, with a stated degree of probability The confidence interval may be expressed using ± notation, such as, 54% ± 3.5% or ranging from 50.5% to 57.5% x ± margin of error

85 Confidence Level The likelihood that the result for the true population lies within the range of the confidence interval Surveys and other studies usually use a confidence interval of 95%, although 90% or 99% is sometimes used

86 Confidence Intervals, Confidence Levels, and Margin of Error When using survey statistics, we must take into account the accuracy of the results. Typically, surveys are conducted amongst a sample of data, rather than the entire population. Therefore, we must consider the following two rules

87 Rule #1: If the sample size stays the same, the margin of error increases as the confidence level increases. *In other words, if your margin of error is higher, you can be more confident in your results Ex) If you have a margin of error of 20% (rather than a margin of error of 5%) your confidence level of your results will increase.

88 Rule #2: If the confidence level stays the same, the margin of error decreases as the sample size increases. *In other words, the more people you survey the more accurate your results Ex) If you survey 100 people (rather than only 10 people) your margin of error will be lower.

89 Example: A telephone survey of 500 randomly selected people was conducted in an urban area. The survey determined that 35% of Canadians (between years old) have an electronic book reader. The results are accurate within ±5%, 19 times out of 20. The number of people in this age range in Canada in 2011 was approx It is assumed that the ages are normally distributed. (workbook p. 130 including solution)

90 Example: How many people in Canada (between years old) would you expect to own an electronic book reader, based on the survey results above? Determine the certainty of the results. Step 1: Determine the confidence interval. Step 2: Determine the number of people who own an electronic book reader. Step 3: Determine the confidence level.

91 Solution Step 1: Determine the confidence interval. x ± margin of error 35% ± 5% Therefore, the confidence interval is 30% to 40%.

92 Step 2: Determine the number of people who own an electronic book reader. Since, the confidence interval is 30% to 40%, and using the approx. population of people, we can determine how many people own electronic book readers = people = people Therefore, to Canadians (between years old) own an electronic book reader.

93 Step 3: Determine the confidence level. The survey results are accurate 19 times out of % = 95% Therefore, the confidence level is 95%.

94 Example: How many people in Canada (between years old) would you expect to own an electronic book reader, based on the survey results above? Determine the certainty of the results. Final statement: It can be said with 95% confidence that between and Canadians (from ages 24-45) have an electronic book reader.

95 Statistical Reasoning Confidence Intervals Assignment: Workbook p #1-6 Due: At the end of the class *Please show work on loose leaf paper. Keep work neat and provide answers in sentence statements. OVERDUE ASSIGNMENTS: p #1-6 p #1-5 p #1-8 p #1-3 p #1-5