1. Statistics What is statistics? Where are statistics used?

2. What is Statistics? Mathematics science pertaining to the collection, analysis, interpretation or explanation, and presentation of data Summarizing information to aid understanding Drawing conclusions from data Estimating the present or predicting the future Random event/process/variable Outcome cannot be predicted Ex: sum of the numbers on two rolled dice Ex: stock market Ex: fluctuations in water pressure in a municipal water supply http://www.scc.ms.unimelb.edu.au/whatisstatistics/

3. Uncertainty in Measurements Where in engineering do we have to worry about uncertainty? Measurements. Ask 50 people to measure the length of a football field with a meter stick… Would each person get the same answer? Why or why not? Field doesn’t change in size; therefore, errors are in the measurements (misreading tick marks, didn’t measure in a straight line, etc.)

4. Uncertainty in Measurements So, how long is the football field? How confident are we that the length is the average value?  Estimate the length by taking the average.  Calculate the standard deviation about the average.

5. Mean, Median, and Mode Mean: the sum of all a list divided by the number of items in the list 1, 2, 4, 5, 5, 8, 9, 12, 15 1, 2, 4, 5, 5, 7, 8, 9, 12, 15 Median: the number separating the higher half of a set of numbers from the lower half Mode: the value that occurs the most frequently in a set of numbers

6. When do you use… Median Mode When you know that a distribution is skewed When you believe that a distribution might be skewed When you have a small number of objects Why? To combat the effect of outliers. When there are many numbers and the frequency of the numbers progress smoothly When you have non-numerical data (categorical data)

7. Standard Deviation and Variance Measures of how spread out a distribution is i.e. Measures of variability Variance: average squared deviation of each number from its mean Summation Standard Deviation: square root of the variance Most commonly used measure of spread

8. Definitions Event/realization Rolling of a pair of dice, taking of a measurement, performing an experiment Outcome Result of rolling the dice, taking the measurement, etc. Deterministic event Event whose outcome can be predicted realization after realization Ex: measured length of a table to the nearest cm Probability Estimate of the likelihood that a random event will product a certain outcome

9. Probability Distribution Functions If a random event is repeated many times  Produces a distribution of possible outcomes, f(n)  The probability distribution function represents the distribution as the percentage of occurrences of each outcome Consider rolling a die. Each side is equally likely to land face up: (Uniform Distribution: f(1) = f(2) = f(3) = f(4) = f(5) = f(6) = 1/6) Distribution function Outcome

10. Probability Distribution Functions Consider rolling a pair of dice: If outcome of this event is the sum of the dice, what does the distribution look like? 6 x 6 = 36 possible ways for the pair of dice to land. 11 possible outcomes (pairs can sum to 2  12). For example, n = 2: (1,1); therefore f(n) = 1/36. n = 7: (1,6) (2,5) (3,4) (4,3), (5,2), (6,1) ; therefore f(n) = 6/36.

11. Probability Distribution Functions Most well-known: bell-shaped distribution (More formally, Gaussian or normal distribution).

12. Normal Distribution Continuous probability distribution Distributions with the same general shape Symmetric with scores more concentrated in the middle than in the tails “Bell shaped” Height is determined by the mean and the standard deviation

13. Probability Distribution Functions f(x)

14. Why are Normal Distributions Important? 1) Many psychological and educational variables are distributed approximately normally - Reading ability, introversion, job satisfaction, etc 2) It’s easy for mathematical statisticians to work with 3) If mean and standard deviations of a normal distribution are known, it is easy to convert back and forth from raw scores to percentiles

15. Standard Normal Distribution Normal distribution with a mean of 0 and a standard deviation of 1. Normal distributions can be transformed to standard normal distributions by: This determines how many standard deviations above or below the mean a particular score is. Ex: Score of 70 on a test with a mean of 50 and standard deviation of 10. Z = 2. (2 standard deviations above the mean) http://www.oswego.edu/~srp/stats/z.htm

17. Percentiles If the mean and standard deviation of a normal distribution are known, you can find out the percentile rank of a person obtaining a specific score Ex: Introductory Psychology test normally distributed with a mean of 80 and a standard deviation of 5 What is the percentile rank of a person with a score of 70? The proportion of the area below 70 is equal to the proportion of the scores below 70 z = (70-80)/5 = -2 f(x)

18. Percentiles What about a person scoring 75 on the test? The proportion of the area below 75 is the same as the proportion of scores below 75. z = (75-80)/5 = -1

19. Percentiles What about a person scoring 90 on the test? This graph shows that most people scored below 90. z = (90 - 80)/5 = 2

20. Percentiles The test has a mean of 80 and a standard deviation of 5. What score do you need to be in the 75th percentile?

22. Percentiles The test has a mean of 80 and a standard deviation of 5. What score do you need to be in the 75th percentile? Look at the z table to find the z associated with 0.75 - The value of z is 0.674 - This means that you need to be.674 standard deviations above the mean Standard deviation is 5, so (5)(.674) = 3.37 Need to be 3.37 points above the mean 80+3.37 = 83.37 Round to get a score of 83

23. Sampling Distribution Say you compute the mean of a sample of 10 numbers; the value you obtain will not equal the population(entire set) mean exactly Sampling distribution of the mean is a theoretical distribution that is approached as the number of samples increases Population with mean of  and standard deviation of , the sampling distribution of the mean has mean of  and standard deviation (called standard error of the mean): where n is the sample size