Statistical Literacy & Data Analysis

Transcription

1 Statistical Literacy & Data Analysis Key Ideas: Quartiles & percentiles Population vs. Sample Analyzing bias in surveys Polls, census & Indices Jan 13 8:43 PM Bell Work 1. find the mean, median and mode for the following set of data. 3, 6, 9, 13, 19, 30, 34, 39 The mean, median, and mode are measure of central tendency for a data set. They represent a typical value for the set. Mean: Median: ( ) 8 = , 6, 9, 13, 19, 30, 34, 39 (13+19) 2 = 16 Mode: The most frequent. In this case their is no mode. Oct 25 8:09 PM 1

2 4.1 Interpreting Statistics Television, radio, newspapers, and Web sites often report statistical data. To understand these reports, you need to be familiar with the statistical language they use. Percentiles: A percentile tells approximately what percent of the data are less than a particular data value. Percentiles are a good way to rank data when you have a lot of data or want to keep data private. Percent vs. Percentile: If you achieved a 72% on a math test it means that you got 72% of the questions correct. If you are in the 72 nd percentile on a math test it means that you got a better grade than 71% of the class. Jan 17 3:05 PM Quartiles A quartile is any of three numbers that separate a sorted data set into four equal parts. The second quartile is the median. It cuts the data set in half. So, it is the same as the 50th percentile. The first, or lowest, quartile is the median of the data values less than the second quartile. It separates the lowest 25% of the data set. So, it is the same as the 25th percentile. The third, or upper, quartile is the median of the data values greater than the second quartile. It separates the highest 25% of the data set. So, it is the same as the 75th percentile. Step 1: Line up the data from smallest to biggest Step 2: Count to make sure you haven't missed any. Step 3: Calculate the 2 nd quartile. Jan 17 3:16 PM 2

3 Example: Calculating Quartiles & Percentiles Here are the hourly pay rates, in dollars, for 17 high school students with part time jobs. a) What are the quartiles for this data set? st = ( ) 2 = nd = rd = ( ) 2 = b) Jimmy got paid $8.00. What percentile does he lie in? Percentile = number of scores below X + 0.5(number of scores equal to X) x 100 total number of scores Percentile = (2) 17 x 100 = 18 Susan lies in the 18th percentile. Jan 17 3:19 PM c) Damien s pay is in the 85th percentile for this group. What is Damien s hourly pay rate? The 85th percentile means approximately 85% of the students in the group earn less money per hour than Damien = = the number of pieces of data Round down to the nearest whole number to determine the number of students who earn less money per hour than Damien: 14 Damien is the 15th student in the ordered list. He earns $11.50 per hour. Oct 25 8:27 PM 3

4 Data Reliability When you read statistical data, you need to think about the reliability of the source. Data from a government agency are usually more reliable than data from someone who is trying to sell a product or promote a point of view. When assessing the validity of survey consider the following 3 factors: Sample size; population size The method of selecting respondents The survey questions Jan 17 3:32 PM Example: Comparing Data Sources In each case, a research topic and two sources of information are described. Decide which data source is more likely to provide reliable data. Oct 25 8:43 PM 4

5 Practice: Page 193 Q. 1a Page Q.1,3,4,9,11,12a,14 Read stats in Q. 6 Oct 25 8:50 PM Bell Work Calculate the Quartiles for the following data: 12, 40, 8, 46, 38, 2, 25, 29, 30, 32, 50, 39 Solution: rd 1 st 2 nd (12+25) 2 = 24 (30+32) 2 = 31 (39+40) 2 = 39.5 Oct 25 8:52 PM 5

6 4.2 Surveys and Questionnaires Population vs. Sample census The population of any set is all the objects in the set. Collecting data about every individual in a population is called a. Conducting a census can be costly and time consuming. It may even be physically impossible. Usually, data are collected for a smaller set of individuals/items selected from the population. This is called a. sample Jan 17 3:44 PM There are 5 key points we must consider when analyzing the validity of a survey: Sample Size Sample size can affect survey results. If the sample is too small, the survey results may not be reliable. If it is too great, the survey may be costly and difficult to administer. Representative Samples A sample needs to be typical of the entire population. This is called a representative sample. If the sample is not representative, it is biased and the survey results are invalid. Sampling Technique Some sampling techniques are random, which means each member of the population has the same chance of being selected. A non random technique may not yield a representative sample. Random techniques Simple random sampling Stratified sampling Cluster sampling Systematic sampling Non random techniques Convenience sampling Judgement sampling Voluntary sampling Jan 17 3:58 PM 6

7 Example 1: Assessing the Sample A town has a population of 20,000 people. The town council conducts a vote at a public meeting about construction a new ice hockey rink. 50 people attend the meeting. 40 of the people at the meeting vote in favour of the hockey rink. The council decides to build the hockey rink since 80% of the people support the idea. a) What percent of the people at the meeting voted for the rink? b) What percent of the people in the town attended the meeting? c) Is the sample representative? Justify your answer. Oct 26 3:46 PM Other important factors which may affect the validity of a survey... Biased Questions Biased questions restrict people s choices unnecessarily or use words that could influence people to answer in a certain way. For results to be valid, survey questions must be unbiased. Survey Techniques Another factor to consider is how the survey is conducted. This is particularly important if any of the questions are about sensitive subjects. People may be more likely to answer honestly if they can reply anonymously in writing rather than responding to an interviewer in person or over the phone. Jan 17 4:00 PM 7

8 To summarize, to assess the validity of a survey, ask yourself these questions: Is the sample size large enough? Is the sample representative? Are the survey questions unbiased? Was the collection method appropriate? Jan 17 4:05 PM Example 2: Assessing the Question Solution: Oct 25 9:16 PM 8

9 Example 3: Assessing the Entire Process Oct 26 3:57 PM Solution: Oct 26 3:58 PM 9

10 Practice: p #2 6,9,10,13 Apr 2 8:51 AM Bell Work Oct 26 4:02 PM 10

11 4.3 The Use and Misuse of Statistics Assessing statistical data Oct 26 4:03 PM Example 1: Assessing Graphs Oct 26 4:06 PM 11

12 Solution: Oct 26 4:07 PM Example 2: Assessing How Data Were Collected and Graphed Oct 26 4:08 PM 12

13 Solution: Oct 26 4:10 PM Practice: p #1 4,7,9 12 Apr 2 8:53 AM 13

14 Bell Work Test scores for a grade 12 math class: 42, 85, 78, 46, 57, 62, 23, 99, 91, 61, 50, 78 23, 42, 46, 50, 57, 61, 62, 78, 78, 85, 91, 99 Jamie got a grade of 57% on his test, what percentile does Jamie lie in? 4/12 x 100 = 25% Jamie lies in the 25th percentile Justin was in the 75th percentile, what percent did Justin get on his test? 0.75 x 12 = 9 Justin had the 10th best mark in the class, so he got 85%. Oct 31 8:29 AM Work with a partner to complete the following investigation. Apr 2 8:54 AM 14

15 4.4 Understanding Indices Price indices help citizen, businesses and industries follow and predict trends in prices. A price index describe the price of an item compared to a base value measured at a particular time or in a particular place. Statistics Canada tracks price changes using several different indices. The most important is the Consumer Price Index (CPI). To determine the CPI, Statistics Canada collects thousands of price quotations from across the country for a basket of about 600 popular consumer goods and services. The items range from French fries and bus fares to tuition and Internet service. Oct 31 8:29 AM Example 1: Reading the Consumer Price Index a) What is the base year for the CPI? Look for the year with CPI of 100. The base year was b) In what year was the cost of the basket of good about 90% of the base cost? When the cost of a basket of good in 90% of the cost in the base year, the CPI will be 90. The CPI was 90 in c) What was the CPI in 1990? What does this mean? In 1990, the CPI was about 78. So, price in 1990 were about 78% of the price in d) Describe the change in the CPI from 1990 to What do you notice about the line segment representing this period? The CPI increased from about 78 to 83. This is an increase of 5% of the base value in one year. This is the greatest one year increase. The line segment representing this increase is the steepest on the graph. e) Describe the overall trend in the CPI and its significance. The CPI increases over the years shown. So, Canadians spend more money each year to buy the same basket of products and services. Oct 31 8:36 AM 15

16 Example 2: Solving Problems Using an Index a) Calculate the average annual rate of inflation from 1990 to b) Use your answer to part a to predict the CPI for Justify your prediction. Oct 31 8:50 AM Some price indices do not show a cane over time. Instead, they compare prices among different geographical regions. Example 3: Using an Index to Compare Cities The 2006 UBS Prices an Earnings report includes a comparison of clothing prices in 71 cities. the base price is the price in New York. a) Which cities in this table have index values less than 100? what does this tell you? b) How do clothing prices in Zurich and Hong Kong compare to clothing prices in New York? Oct 31 8:55 AM 16

17 Practice: p.194 #1 2 p #1 5, 9 12 Note: Monday Assignment due on Tuesday Test Next WEDNESDAY Apr 2 8:57 AM 17