Statistics Math 416

Game Plan Introduction Introduction Census / Poll / Survey Census / Poll / Survey Population – Sample – Bias Population – Sample – Bias Sample Proportion Sample Proportion Mean Median Mode Mean Median Mode Box and Whisker Plot Box and Whisker Plot Box and Whisker Interpretation Box and Whisker Interpretation

Stats Intro There are lies, there are damn lies and then there are statistics There are lies, there are damn lies and then there are statistics - Mark Twain - Mark Twain The goal is by the use of number describe a characteristic of a population. The goal is by the use of number describe a characteristic of a population. The idea is to win your argument by providing facts and too many people consider statistics to be absolute facts. The idea is to win your argument by providing facts and too many people consider statistics to be absolute facts.

Stats Intro In general, most people do not understand statistics. In general, most people do not understand statistics. Hypothesis: Student A has a school average of 10% Hypothesis: Student A has a school average of 10% Conclusion: Student A is a bad person. Conclusion: Student A is a bad person. The statistic does not measure the person’s goodness or badness. The statistic does not measure the person’s goodness or badness. What does that statistic mean? What does that statistic mean? If all there marks were the same for all courses, it would be 10% If all there marks were the same for all courses, it would be 10%

Statistics Life is a continual battle to get your ideas across and have other people trying to get their ideas across to you. Life is a continual battle to get your ideas across and have other people trying to get their ideas across to you. You are constantly being bombarded by arguments and statistics. You are constantly being bombarded by arguments and statistics. 1. Commercials 2. Teachers To understand the world around you, need to be aware of statistics meaning and reliability. To understand the world around you, need to be aware of statistics meaning and reliability. Where do statistics come from? Where do statistics come from?

Population First we establish the population. First we establish the population. Population: the complete group that we are investigating Population: the complete group that we are investigating Characteristic: A particular identifying object exhibited by the population Characteristic: A particular identifying object exhibited by the population i.e. hair colour favorite colour math knowledge, political opinion etc.

Population The next problem is interpreting how to measure a characteristic and obtain the data. The next problem is interpreting how to measure a characteristic and obtain the data. Obtaining the Data: Three methods Obtaining the Data: Three methods Method #1: Ask the whole population - Called a census - Problems – hard to do – depending on population

Census Method #1: Ask the whole population Method #1: Ask the whole population - Called a census - Problems – hard to do – depending on population

Poll Method #2: - Ask a representative “sample” of the population Method #2: - Ask a representative “sample” of the population - Called a poll Problems: representative may be tricky

Survey Method #3: Ask only experts of the population Method #3: Ask only experts of the population - called a survey Problems: who is an expert Representative sample?

Bias Bias If data is obtained or presented in an unfair manner than all conclusions are not correct. The results are said to be biased (or unfair). If data is obtained or presented in an unfair manner than all conclusions are not correct. The results are said to be biased (or unfair). In collection How and who you ask is the main source of bias How and who you ask is the main source of bias There are 4 types of bias (bad sampling, non pertinence, wording of question & attitude of pollster). There are 4 types of bias (bad sampling, non pertinence, wording of question & attitude of pollster).

Bias Eg asking 5 yr olds their favorite beer Eg asking 5 yr olds their favorite beer Bad sampling Bad sampling Eg Do you like to play an instrument? (to find favorite color) Non-pertinence Non-pertinence Eg man I hate Bush, are you in favor of war? Wording of questions Wording of questions Eg a policeman asking were you speeding? Attitude of pollster Attitude of pollster

Presentation Bias In presentation, imagine you disregard a grade level and claim that they do not matter in a school’s decision. In presentation, imagine you disregard a grade level and claim that they do not matter in a school’s decision. I need to prove my product is the best, how can I get these numbers to show that? I need to prove my product is the best, how can I get these numbers to show that?

Buy This Stock! $400 $300 $0 $100 $200 $500 JanAprilMarchFebJan Not! A statistical presentation is always biased Stencil #1-3

Representative Sample Creating a representative sample can be an art form in itself. The sample should be in all the same proportions, an impossibility. Creating a representative sample can be an art form in itself. The sample should be in all the same proportions, an impossibility. You must focus on the characteristics (the poll or survey is focusing on!) You must focus on the characteristics (the poll or survey is focusing on!)

Representative Sample Consider a school has 50 boys and 25 girls and a representative sample of 10 needs to be created. Consider a school has 50 boys and 25 girls and a representative sample of 10 needs to be created. We note the population is described in terms of boys and girls hence we will need to create our sample on that basis We note the population is described in terms of boys and girls hence we will need to create our sample on that basis Three steps Three steps

Representative Sample 1) Relative (by percent) 50/75 = 67% 25/75 = 33% n = 75 2) Theory - sample = 10 10x.67=6.7 10x.33 = 3.3 Difficult to get.7 or.3 of a person! 3) Reality 7 3 Total of 10 & has added bias

Some Rules If it starts at zero it stays at zero If it starts at zero it stays at zero If it appears to be zero be careful! If it appears to be zero be careful! Make decisions on a category not overall Make decisions on a category not overall

Creating a Sample 1) Given the following, create a sample of 10 Hudson Non-Hudson Hudson Non-Hudson Young n = 109 Relative Middle Aged Old MA 18% 22% 0 28% 30% Y0 0Is it really 0 people?

Creating a Sample Y open here Reality MA O Y 0 0 Theory MA O Stencil 4,5, 6 Do relative, theory and reality for #4; in #5 & 6 put theory & relative together

Statistics Central Tendency - Mean Mean means Mean meansthe average Symbol x Found by dividing the sum ∑x i by the number of elements n. i.e. x = ∑ x i Means which value would all values be equal to if they were the same i.e. (5,9,3,6) x = ∑ x i n n = ( )/4 = 5.75

Mode Symbol M Symbol M It is the number that appears the most It is the number that appears the most It is possible, not to have any or to have more than one mode It is possible, not to have any or to have more than one mode Eg (1,2,5) Eg (1,2,5) Eg (1,6,6,8) Eg (1,6,6,8) Eg (1,3,3,4,4,8) Eg (1,3,3,4,4,8) M = (nothing repeats) M = 3 & 4 M = 6

Median Symbol M Symbol M Median is found as the middle value Median is found as the middle value Note the sample must be in order! Note the sample must be in order! There are two possibilities (odd & even) There are two possibilities (odd & even) Consider (1,5,7) n = 3 Consider (1,5,7) n = 3 Odd only 1 middle; M = 5 Odd only 1 middle; M = 5 (1,5,7,8) n = 4 (1,5,7,8) n = 4 You must find the mean of both middles (5 + 7)/2 = 6 You must find the mean of both middles (5 + 7)/2 = 6 Do #7

Box & Whisker Plot (2,5,1,6,9,8,) (2,5,1,6,9,8,) The Construction The Construction 1) Make sure your sample is in order 1) Make sure your sample is in order (1, 2,5,6,8,9) (1, 2,5,6,8,9) 2) Find the min, max & median 2) Find the min, max & median Min = 1 ; max = 9 median = 5.5 = Q 2 Min = 1 ; max = 9 median = 5.5 = Q 2 These three points will serve you as part of the box and whisker diagram. Draw it on board… These three points will serve you as part of the box and whisker diagram. Draw it on board…

Box & Whisker Plot ) Create a number line with vertical line at the 3) Create a number line with vertical line at the three points hinges three points hinges 4) Find median between min and Q 2 called Q1 4) Find median between min and Q 2 called Q1 5) Find median between Q2 and max called Q3 It is 2 and make another hinge It is 2 and make another hinge It is 8 & make another hinge. (1,2,5,6,8,9) Complete it!

Words & Facts We have broken the data into four parts called quartiles. We have broken the data into four parts called quartiles. Whiskers Min Box Whiskers Q1Q1 Q3Q3 Q2Q2 Interquartile range = Q3-Q1 Max

Words & Facts Each quartile should hold about ¼ of the data BUT you cannot be sure Each quartile should hold about ¼ of the data BUT you cannot be sure You cannot tell the mean or the mode You cannot tell the mean or the mode Do not jump to conclusions! Do not jump to conclusions! A box and whisker gives you an idea about the spread or concentration or dispersion of data A box and whisker gives you an idea about the spread or concentration or dispersion of data

Example # A general View A general View This data is very close together below 4. There is more of a spread between 4 and 11 and once again between 11 and 12. This data is very close together below 4. There is more of a spread between 4 and 11 and once again between 11 and 12. Some Questions… What is the Mean? 12 No idea

Questions What is the mode? What is the mode? What is the median? What is the median? What is the interquartile range? What is the interquartile range? How many are below 11? How many are below 11? No idea Q 2 = 4 Q 3 -Q 1 = = 8 75% but no idea of the number The lowest concentration of numbers lie where? Lowest concentration vs. highest concentration Between

Example # n = 20 Class A n = 40 Class B a) Which class did better?Hard to tell but class A b) What are the meansNo idea ¾ x /4 x 40 = 35 c) All together approximately how many were over 60%

Example #2 – More Questions Which class and which mark was the highest? Which class and which mark was the highest? Class B at approximately 97% Class B at approximately 97% Which class has lowest range? Which class has lowest range? Class A Class A = = 32 Class B Class B = = 57 Answer: Class A Answer: Class A Finish Stencil Finish Stencil