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# Statistics Review

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Brief Review of Market Research concepts for Market Research

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• Parameter- a measurement of of the Parent Population.(e.g. average income of all the Volvo in the world)
• NONProbabuility Samples:Convenience Samples Jugement Samples --sample elements are hand-picked which is what we doQuota Samples-
• Often we don’t know how the “real world” population is distributed, so we have to estimate ().There is no problem for two reasons: 1. variation usually changes for most variables of interest in marketing So, if the study is a repeat we can use old values we found for .2. We can calculate smaple variance Need to add blurb on “standard error of estimate”
• Often we don’t know how the “real world” population is distributed, so we have to estimate ().There is no problem for two reasons: 1. variation usually changes for most variables of interest in marketing So, if the study is a repeat we can use old values we found for .2. We can calculate smaple variance Need to add blurb on “standard error of estimate”
• Often we don’t know how the “real world” population is distributed, so we have to estimate ().There is no problem for two reasons: 1. variation usually changes for most variables of interest in marketing So, if the study is a repeat we can use old values we found for .2. We can calculate smaple variance Need to add blurb on “standard error of estimate”
• Often we don’t know how the “real world” population is distributed, so we have to estimate ().There is no problem for two reasons: 1. variation usually changes for most variables of interest in marketing So, if the study is a repeat we can use old values we found for .2. We can calculate smaple variance Need to add blurb on “standard error of estimate”
• Often we don’t know how the “real world” population is distributed, so we have to estimate ().There is no problem for two reasons: 1. variation usually changes for most variables of interest in marketing So, if the study is a repeat we can use old values we found for .2. We can calculate smaple variance Need to add blurb on “standard error of estimate”
• Often we don’t know how the “real world” population is distributed, so we have to estimate ().There is no problem for two reasons: 1. variation usually changes for most variables of interest in marketing So, if the study is a repeat we can use old values we found for .2. We can calculate smaple variance Need to add blurb on “standard error of estimate”
• Revenues above are from Wholesale Carrier B&C Forecast.From Frost & sullivan report 2006
• ### Statistics Review

1. 1. Statistics for Market Research A Brief Refresher Course by Brian Neill
2. 2. Contents s Quick Review of Basic Concepts in Statistics: – Mode, Median, Mean, Sampling, Normal Curve s Intermediate Concepts – Confidence Intervals, Probability Sampling s Formulas & Calculations ________________________ s CAGR explained 2
3. 3. Purpose s To provide a brief refresher of the basics of statistics as it pertains to Market Research. Geared towards those with previous training in statistics. 3
4. 4. Statistics Basic Concepts 4
5. 5. Definitions & Basic Concepts s Mean - the average of all the data points. s Mode - the most common data point to occur – (e.g. \$4.99 might be most common price at Wal-Mart) s Median - the middle number in a ranking of data points (e.g. 7 students ranked from tallest to shortest, 4th one’s height is the median) s Parent Population - the totality of cases being studied. (e.g. All the Volvo owners in the world). This is the group of people or things we are trying to find out about. s Its not practical to measure the entire parent population, so Sampling is used to reflect the Parent Population. (e.g. We interview a sample of Volvo owners ) 5
6. 6. Sampling used to measure Parent Population ******* >>>^^^^^^^ ********^^^^ >***>^ ^^*******>>>> ^^^**^ ********^^^^^^ ^^^^^^^^^^^ Sample is taken from parent population. Measurements ^^^>>>>>>> are taken on the sample. If ******^^ sampling was done correctly. measurements are * representative of the Parent Population 6
7. 7. Definitions & Basic Concepts s Sampling Distribution - If we sample correctly, we can generalize our findings to the “real world” population we are studying. s e.g. We take a sample of Volvo owners. We ask them how many miles they drove in 1999. (Their answers are a Statistic - a measurement that we perform on the sample) s If we did the sampling correctly, we can make inferences about ALL Volvo drivers (The Parent Population) 7
8. 8. Definitions & Basic Concepts s Probability samples – each population element has a known chance of being included in the sample. We can then make inferences about the Parent Population. s Non-probability samples – samples using personal judgement. Since selection is non-random, there is no way of estimating the probability that any element is sampled. – Cannot estimate the adequacy of the sample result. Therefore cannot be sure your data reflects the entire population. 8
9. 9. Definitions & Basic Concepts the Normal Distribution s Central Limit Theorem: Take a big enough random sample (probability sample) and your sample distribution will look like a normal curve. – you can calculate the mean and the variance, and confidence limits. – You can make inferences about the Parent Population. To make inferences using statistics your sample must be “large enough” and it must be random 9
10. 10. Definitions & Basic Concepts s How Large a Random Sample do you need? s Answer: Depends on how much variation you see in the group you are studying. – If a lot of variation, you will need a larger sample size. – If little variation, need smaller sample size. THE SIZE OF THE PARENT – POPULATION HAS NO DIRECT Although, larger populations usually have larger variations, so size hasSIZE OF SAMPLE EFFECT ON indirect effect at times. NEEDED! 10
11. 11. Definitions & Basic Concepts s Example of Little Variation: Avg.Diameter of U.S. Dime Coins. These coins are made to exacting standards, so even though there are many millions of dimes (i.e.. large parent population), you would need only a small sample to test diameter. s Example of Wide Variation: Average Height of City of Dallas’ 8,200 employees. Much greater variance than size of US dimes, so a bigger random sample is needed to test. s Example of Wider Variation. Average height of City of Dallas employees and their children. Even greater variance, so even bigger sample needed. 11
12. 12. Practical Example Pop Quiz: In a Concept test we talk to 10 Companies. We find that they would spend an average of \$1 million on product x this year. Q: Can we conclude that Companies in U.S. would, on average, spend \$1 million on product x? Nein! (NO!) 1) Sample not large enough (10 Companies is not enough, unless there is little variation among Company attitudes, which seems unlikely). 2) Plus your sample of respondents not chosen at random. 12
13. 13. Intermediate Concepts Onward and Upward 13
14. 14. Confidence Intervals s If sample is large and randomly selected, it looks like a normal curve. Normal curves have helpful properties... s In any Normal curve, 95 percent of the values are within ± 1.96 Standard Deviations (σ) of the mean. • 68.26 percent of the sample means will be within ± 1.0 (σ ) of the population mean. Note: S is our estimate of the standard deviation of the parent population (σ ) is the actual standard deviation 14
15. 15. Confidence Intervals The Normal Curve 16 14 12 10 8 6 4 1 std 2 std 2 3 std deviation deviations 0 deviations -3 -2 -1 0 1 2 1.96 std deviations S (σ) = 95% of all observations 15
16. 16. Example of Confidence Intervals s Let’s say we wanted to find the average number of miles a Volvo is driven in a year (to plan our new warranty) s We randomly sample a large group of Volvo drivers and find that the average number of miles driven is 17,000 miles. • (But their was a considerable variance) – We choose a range ± 1.96S ± 3,000 miles. This means we can be 95% sure that the TRUE average number of miles is within plus or minus 3,000 miles of 17,000 – So are 95% confidence interval is -> 14,000 to 20,000 miles per year. 16
17. 17. Confidence Intervals 95% chance the The Normal Curve true mean falls 16 within here 14 12 10 8 6 4 2 1 std 2 std 0 deviation deviations -3 -2 -1 0 1 2 14,000 Est. Mean = 17,000 20,000 1.96 std deviations S (σ) = 95% of all observations 17
18. 18. Example of Confidence Intervals s Suppose we decide this is too big of a spread. So we will accept less certainty in order to narrow our range s So we decide to go with 68.2 % of all owners (1 std deviation away from the average) =1,500 miles s Then we can say that there is a 68.2% chance of a Volvo being driven 15,500 to 18,500 miles per year. 18
19. 19. Confidence Intervals The Normal Curve 68% chance the true mean falls within here 16 14 12 10 8 6 1 std 2 std 4 deviation deviations 2 0 -3 -2 -1 0 1 2 15,500 17,000 18,500 1 S (σ) std deviations = 68.2% of all observations 19
20. 20. Example of Confidence Intervals s Alternative: s If the Marketing department at Volvo did not want to sacrifice precision and confidence, they could go out and re-test using a much larger sample of respondents. s Remember, where there is a lot of variance, you need a bigger random sample to make up for it. 20
21. 21. Formulas & Calculations “oooh me brain hurts” Mr. Gumby—character on “Monty Python’s Flying Circus” 21
22. 22. Finding Confidence Intervals s When Population Variance unknown s First find sample variance Š, by / Š²= (X- Avg.) ² (n-1) s Then find Standard error of estimate of the mean S = Š/n s for 95% confidence, 1.96 standard deviations, S s so limits equal: Avergage-1.96S and Average + 1.96S 22
23. 23. Sample Size - When finding an average or mean s When variance unknown, use n = (z ²/r ²) * C² s where z =confidence interval in std deviations s r = relative precision desired s C = your estimate of the variation of the sample (one deviation away from the mean equals how many miles) 23
24. 24. Sample Size - When finding an average or mean s Example want to find average miles driven by Volvo owners to plus/minus 500 miles, want 95% confidence in the results, and we think 1 std deviation will equal 3000 miles s n=(2²/500²)*3000² s n = 144 s we need a sample of 144 to get this level of accuracy in estimating the average miles 24
25. 25. Sample Size - When finding a proportion s When variance is unknown s n = (z ²/r ²) * Π(1-Π) s where s z=confidence interval in std deviations s r = relative precision desired s Π = this is not “pi”. It’s your estimate of the proportion of the population that has the characteristic you are looking for (e.g.. Percent of Companies in our sales area who are interest in a product). That’s right, you must estimate the very quantity you are trying to find, in order to determine sample size! 25
26. 26. Sample Size - When finding a proportion s For example, if we want to find how many Companies in our Sales area will want to buy product Q. We want 95 % confidence & a precision of plus/minus 5 percent. We guess that 20% will be interested. s n = (z ²/r ²) * Π(1- Π) s n=(2²/0.05²)* (0.2)(1-0.2 ) s n = 256 s we must survey a sample of at least 256 Companies to get this level of accuracy in estimating the proportion that want product Q. 26
27. 27. CAGR Explained Measuring Growth Rate 27
28. 28. Measuring Growth Rate Compound Annual Growth Rate (CAGR) Compound Annual Growth Rate is the most common measure of growth in most industries. CAGR removes the compounding portion to show a more accurate picture of growth (removes “the interest on interest”). Example Example: Let’s calculate the Year Revenues average annual growth (\$Billion) 1997 1.28 1997-2005. 1998 1.57 Using two methods: 1999 1.78 2000 2.01 1) Arithmetic (wrong) way 2001 2.22 2) CAGR 2002 2.43 2003 2.64 2004 2.84 2005 3.04 28
29. 29. A Common mistake when calculating the Growth Rate s Your first instinct might be to just take an “average” of growth and divide by number of time periods- but that will yield a wrong answer! = (YRn-YR1)/ YR1 Where: n •YR1 = Year 1 (first year) revenues •YRn= Year n revenues (the last year of the =(3.04 -1.28)/1.28 forecast period) 8 years •n = number of years used for growth rate (count forward = 17% avg. growth rate from YR1 to YRn, beginning year to ending year) You end up with an answer that is too large because you have not removed the compounding portion (“the interest on interest”). 29
30. 30. Solution – The correct way to calculate CAGR in our example: CAGR= ((YRn/YR1)1/n ) - 1 •Where: YR1 = Year 1 (first year) revenues YRn= Year n revenues (the last year of the forecast period) n = number of years used for growth rate (count forward from YR1 to YRn, beginning year to ending year) CAGR = (3.04/1.28) 1/8 ) -1 = 1.114 - 1 CAGR = 0.114186 = 11.4186% The correct answer in our example is that revenues grew 11.4% per year. 30
31. 31. Proof that CAGR is right We can check our answer by adding 11.4186% growth every year and you’ll arrive at the correct final year revenue Revenues (\$) 1997 start 1.280 1998 add11.4186% 1.426 1999 add11.4186% 1.589 2000 add11.4186% 1.770 2001 add11.4186% 1.973 2002 add11.4186% 2.198 2003 add11.4186% 2.449 3.04B, 2004 add11.4186% 2.728 exactly 2005 add11.4186% 3.040 right! 31
32. 32. Plotting These Average Growth Rates 17% 32
33. 33. Reference s Churchill, Gilbert, A., Marketing Research, Methodological Foundations, 7th Ed. Dryden Press. Chapters 10 and 11. (Gilbert teaches at U of Wisconsin, Madison) 33
34. 34. The End 34