Date Lesson TOPIC HOMEWORK. Displaying Data WS 6.1. Measures of Central Tendency WS 6.2. Common Distributions WS 6.6. Outliers WS 6.

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1 UNIT 6 ONE VARIABLE STATISTICS Date Lesson TOPIC HOMEWORK Displaying Data WS 6.1 Measures of Central Tendency WS Grouped Data Central Tendency Measures of Spread (I) Box Whisker Plots/ Quartiles Measures of Spread (II) Standard Deviation WS 6.3 WS 6.4 WS Common Distributions WS 6.6 Outliers WS 6.7 Misleading Graphs WS 6.8 Review for Unit 6 Test WS UNIT 6 TEST

2 MBF 3C Lesson 6.1 Displaying Data Terminology Categorical Data data that are types rather than numbers. ie: colours or types of snack foods Continuous Data data that can have any numerical value within a finite or infinite interval. ie: the heights of students in class Discrete Data data that are distinct and can be counted. ie: the number of students who like broccoli BAR GRAPHS and CIRCLE GRAPHS John has a job at a music store. The table shows John s expenses last month. Expense Amount ($) Entertainment 100 Clothing 225 Cell phone 50 Lunch 75 Transportation 80 Rent 150 Create a bar graph to represent the data. y x

3 Complete the table. Calculate the percent and angle measure for each expense. Create a circle graph to represent the data. Include a title and label the sectors. Expense Amount($) Percent (%) Measure of Angle ( ) Entertainment 100 Clothing 225 Cell Phone 50 Lunch 75 Transportation 80 Rent 150

4 Histograms William measured the heights of students in his math class. The heights are rounded to the nearest centimetre Height can have any numerical value, so this is continuous data. Complete the frequency table below. Interval Tally Frequency [ ) [ ) [ ) [ ) [ ) [ ) [ ) A square bracket is used to indicate that the value is included in the interval. A round bracket is used to indicate that the value is not included in the interval. The interval [ ) includes all heights from 130 cm up to but not including 140 cm. Graph the data with the intervals on the horizontal axis and the frequency on the vertical axis. Include a title and label the axes frequecy Students Heights height

5 Interpreting a Bar Graph Colleen surveyed students at her school about their favourite sports. Use her bar graph below to answer the following. a) Which sport is most popular? b) Which sport is least popular? c) Does your answer in (b) mean that students don't like that sport? d) How many students like volleyball most? e) How many students took part in the survey? Analyze Continuous Data The histogram shows the masses of a sample of patients at a hospital. a) How many patients weigh at least 85 kg but less than 90 kg? b) How many have a mass of at least 100 kg? c) How many are in the sample? d) Find the percent of patients who have a mass of at least 100 kg. WS 6.1

6 MBF 3C Lesson 6.2 Measures of Central Tendency Terminology Mean the sum of all values in a set of data divided by the number of values in the set Median the middle value of the set of data when data is ordered from least to greatest Mode the value or attribute that occurs most often in a set of data Outliers a value far away from other values in a set of data Ex. 1 Find the mean, median, and mode of the following test scores. 61, 76, 89, 72, 65, 71, 61, 83, 45, 68, 62, 59, 71, 68, 69, 86 Mean ( a.k.a. average) sum of all terms in the data set Mean = number of terms Median Arrange in order from least to greatest. The middle number(s) give the median. 45, 59, 61, 61, 62, 65, 68, 68, 69, 71, 71, 72, 76, 83, 86, 89 45, 59, 61, 61, 62, 65, 68, 69, 71, 71, 72, 76, 83, 86, 89 Mode the value that occurs the most often. (it is possible to have more than one mode.) 45, 59, 61, 61, 62, 65, 68, 68, 69, 71, 71, 72, 76, 83, 86, 89 WS 6.2

7 MBF3C Lesson 6.3 Grouped Data (measures of central tendency) When many numbers are repeated in a list of data, it often is easier to list the data in grouped form 1. The following table shows the annual salaries earned by employees of a small company. Determine the mean, median, and mode. x Annual Salary ($) SUM: f Frequency Cumulative Frequency f x The above table is a simpler way of listing: 35500, 35500, 35500, 42750, 42750, 42750, 42750, 42750, 51000, 51000, 51000, 51000, 51000, 99000, If the data are grouped into intervals, the calculated values for mean, median, and mode will only be estimates. Calculate the mean, median and mode of these amounts of charitable donations. x Amount of Donation ($) SUM: f Frequency Cumulative Frequency f x

8 3. Use the data on the bar graph to complete the table. Then calculate the mean, median, and mode. x f Frequency Cumulative Frequency f x SUM: WS 6.3

9 MBF 3C Lesson 6.4 Measures of Spread (I) Terminology Quartiles three values that divide a set of data into four equal intervals with equal numbers of data Range the difference between the greatest and least values in a set of data Box and whisker plot a graph representing the first quartile, the median, and the third quartile of a set of data with a box the least and greatest data are represented by lines (whiskers) extending from the box Interquartile range the range of the central half of data when the data are arranged from least to greatest a measure of how closely the data clusters around the mean Ex. 1 The table below shows the marks from Mr. Jeacle's MCR 3U class last semester a) Find the median, first quartile (Q1) and the third quartile (Q3) and the interquartile range. (i) Arrange from least to greatest

10 b) Display the data in a box and whisker plot Ex. 2 Rebecca works at a computer store. She records the number of computers that she sells each month. For the last 12 months her totals were: 51, 17, 25, 39, 7, 49, 62, 41, 20, 6, 43, 13 Find the median, first quartile (Q1) and the third quartile (Q3) and the interquartile range. 6, 7, 13, 17, 20, 25, 39, 41, 43, 49, 51, 62

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12 MBF 3C Lesson 6.5 Measures of Spread (II) Terminology Variance the mean of the squares of the deviations from the mean of a set of data it is a measurement of how spread out the values in a set of data are from the mean. The greater the variance, the greater the spread of the data. Variance = (x 1 mean) 2 (x 2 mean) 2 (x 3 mean) 2... (x n mean) 2, where x 1, x 2, x 3, are n the values in the data set and n is the number of values in the data set. Standard Deviation the typical distance of a particular value from the mean. The greater the standard deviation, the greater the spread of the data. Standard Deviation ( ) = variance 2 2 ( x x) ( x x) ( x x)... ( x x) 1 2 n 3 2 n 2 This looks complicated, but it is much simpler when we use a table to organize our work. Ex. 1 Anthony's weekly food expenses for the first 10 weeks of the year are as follows. $61, $83, $77, $88, $67, $71, $65, $72, $67, $90 Find the variance and the standard deviation, correct to the nearest cent. Food Expenses x x (mean) Sum x x 2 ( x x) Variance ( x x) n 2 Standard Deviation( ) ( x x) n 2

13 Ex. 2 Find the variance and the standard deviation for each of the following data sets. Set A x Sum x x x 2 ( x x) Set B x Sum x x x 2 ( x x) WS 6.5

14 MBF3C Lesson 6.6 Common Distributions Identify the type of distribution:

15 Quick Review: For the Grade 9 French marks listed above, calculate the mean, median, and mode x f Frequency Cumulative Frequency f x Mean: Median: SUM: Mode: WS 6.6

16 MBF3C Lesson 6.7 Outliers In statistics, an outlier is an observation point that is distant from other observations. An outlier may be due to variability in the measurement or it may indicate experimental error; the latter are sometimes excluded from the data set. 1. An online survey was taken, asking people to estimate how many apples that they ate last month. The data from the survey are as follows: 2, 7, 8, 4, 6, 15, 18, 177, 9, 3 Calculate the statistics and fill in the box at the right. Which measure of central tendency do you trust the most, and why? Mean. Median. In a case like this, the statistician may decide it is best to, and then recalculate the statistics. Range. Standard Mean. Deviation. Median. Variance Range. Standard Deviation. Variance

17 2. Class marks on an assignment are as follows: 75, 45, 80, 90, 0, 80, 0, 85, 0, 0, 70, 90. The principal asks the teacher what the average score was for the class on the assignment. Show at least two different ways that the teacher could report the answer. Mean. Median. Range. Standard Deviation. Variance WS 6.7

18 MBF3C Lesson 6.8 Misleading Graphs What can make a graph misleading? 1. If the scale is too narrow or there is not a big enough range. Data looks like it has a big difference really does not. 2. If there is no scale at all. 3. If the scale is truncated (cut off at the bottom)

19 3. If the bars are not equal widths. 4. If the intervals on the graph are not equal, but are depicted as equal. 5. A pictogram can be misleading if all of the pictured items are not the same size.

20 Ex. What is misleading about the graphs below and what could be done to fix them so that they are no longer misleading? a) b) c) d) WS 6.8

21 DO NOT PRINT ANYTHING AFTER THIS PAGE

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24 The PC Changebook of 2011 was a government pamphlet FILLED with misleading graphs!

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26 MBF 3C Lesson 6.6 Common Distributions Normal Distribution a bell shaped distribution that is symmetrical about the mean - the mean, median, and mode are close in value and are located at the centre of the distribution and any measure of central tendency is representative of the data. Bimodal Distribution a distribution that contains two equally likely measures of central tendency within the data and the mode is most representative of the data. - it has two peaks with frequencies clustering around two sub-groups. Skewed Distribution a non-symmetrical distribution of data - there is a greater cluster of data on either the right or left side. - it has the appearance of a normal distribution pushed to one side. NO measure of central tendency is representative of the data Skewed Right Distribution WS 6.5