Transcript Chapter 1
Chapter 1 The mean, the number of observations, the variance and the standard deviation Some definitions Data - observations, measurements, scores Statistics - a series of rules and methods that can be used to organize and interpret data. Descriptive Statistics - methods to summarize large amounts of data with just a few numbers. Inferential Statistics - mathematical procedures to make statements of a population based on a sample. More Definitions Parameter - a number that summarizes or describes some aspect of a population. Sampling Error - the difference between a statistic and its parameter. Non-parametric Statistics - statistics for observations that are discrete, mutually exclusive, and exhaustive. Where we are going Descriptive Statistics Number of Observations Measures of Central Tendency Measures of Variability Observations Each score is represented by the letter X. The total number of observations is represented by N. Measures of Central Tendency Finding the most typical score median - the middle score mode - the most frequent score mean - the average score Calculating the Mean Greek letters are used to represent population parameters. (mu) is the mathematical symbol for the mean. is the mathematical symbol for summation. Formula - = (X) / N English: To calculate the mean, first add up all the scores, then divide by the number of scores you added up. The mode, the median and the mean Ages of people retiring from Rutgers this year. 60 63 45 63 Mode is 63. 65 70 55 60 65 63 45 55 60 60 63 Median is 63. 63 63 65 65 70 X = 609 N = 10 Mean = 60.90 Measures of Variability Range - the distance from the highest to the lowest score. Inter-quartile Range - the distance from the top 25% to the bottom 25%. Sum of Squares (SS) - the distance of each score from the mean, squared and then summed. Variance (2)- the average squared distance of scores from mu (SS/N) Standard Deviation ()- the square root of the variance. Computing the variance and the standard deviation Scores on a 10 question Psychology quiz Student X John 7 Jennifer 8 Arthur 3 Patrick 5 Marie 7 X = 30 N=5 = 6.00 X- +1.00 +2.00 -3.00 -1.00 +1.00 (X- ) = 0.00 (X - )2 1.00 4.00 9.00 1.00 1.00 (X- )2 = SS = 16.00 2 = SS/N = 3.20 = 3.20 = 1.79 The variance is our most basic and important measure of variability The variance ( =sigma squared) is the average squared distance of individual scores from the population mean. Other indices of variation are derived from the variance. The average unsquared distance of scores from mu is the standard deviation. To find it you compute the square root of the variance. 2 Other measures of variability derived from the variance We can randomly choose scores from a population to form a random sample and then find the mean of such samples. Each score you add to a sample tends to correct the sample mean back toward the population mean, mu. The average squared distance of sample means from the population mean is the variance divided by n, the size of the sample. To find the average unsquared distance of sample means from mu divide the variance by n, then take the square root. The result is called the standard error of the sample mean or, more briefly, the standard error of the mean. We’ll see more of this in Ch. 4. Making predictions (1) Without any other information, the population mean (mu) is the best prediction of each and every person’s score. So you should predict that everyone will score precisely at the population mean. Why? Because the mean is an unbiased predictor or estimate (that is, the deviations around the mean sum to zero). Making predictions (2) The mean is precisely the number that is the smallest squared distance on the average from the other numbers in the distribution. Thus, the mean is your best prediction, because it is a least squares, unbiased predictor. What happens if we make a prediction other than mu. Scores on a Psychology quiz (mu = 6.00) What happens if we predict everyone will score 5.50? Student X John 7 Jennifer 8 Arthur 3 Patrick 5 Marie 7 X = 30 N=5 = 6.00 X X -- 5.5 5.50 +1.50 +2.50 -2.50 -0.50 +1.50 (X- ?) = 2.50 (X(X- 5.50) - )2 2 2.25 6.25 6.25 0.25 2.25 (X- ?)2 = SS = 17.25 2 = SS/N = 3.45 = 3.20 = 1.86 Compare that to predicting that everyone will score right at the mean (mu). Scores on a 10 question Psychology quiz Student X John 7 Jennifer 8 Arthur 3 Patrick 5 Marie 7 X = 30 N=5 = 6.00 X- +1.00 +2.00 -3.00 -1.00 +1.00 (X- ) = 0.00 (X - )2 1.00 4.00 9.00 1.00 1.00 (X- )2 = SS = 16.00 2 = SS/N = 3.20 = 3.20 = 1.79 But when you predict that everyone will score at the mean, you will be wrong. In fact, it is often the case that no one will score precisely at the mean. In statistics, we don’t expect our predictions to be precisely right. We want to make predictions that are wrong in a particular way. We want our predictions to be as close to the high scores as to the low scores in the population. The mean is the only number that is an unbiased predictor, it is the only number around which deviations sum to zero. We want to be wrong by the least amount possible In statistics, we consider error to be the squared distance between a prediction and the actual score. The mean is the least average squared distance from all the scores in the population. The number that is the least average squared distance from the scores in the population is the prediction that is least wrong, the least in error. Thus, saying that everyone will score at the mean (even if no one does!) is the prediction that gives you the smallest amount of error. So the mean is the best prediction of everyones’ score because it is a least squares, unbiased predictor for all the scores in the population. Why doesn’t everyone score right at the mean? Sources of Error Individual differences Measurement problems If we predict that everyone will score right at the mean, how much error do you make on the average? To find out, find the distance of each score from the mean, square that distance and divide by the number of scores to find the average error. WHOOPS: THAT’S SIGMA2. Mean square for error = variance Questions and answers – the mean. WHAT QUALITIES OF THE MEAN (MU) MAKE IT THE BEST PREDICTION YOU CAN MAKE OF WHERE EVERYONE WILL SCORE? The mean is an unbiased predictor or estimate, because the deviations around the mean always sum to zero. The mean is a least squares predictor because it is the smallest squared distance on the average from all the scores in the population. Q & A: the mean WHY WOULD YOU PREDICT THAT EVERYONE WILL SCORE AT THE MEAN WHEN, IN FACT, OFTEN NO ONE CAN POSSIBLY SCORE PRECISELY AT THE MEAN? In statistics, we don’t expect our predictions to be precisely right. We want to make predictions that are close and wrong in a particular way. We want least squares, unbiased predictors. Q & A: The variance WHAT ARE THE OTHER NAMES FOR THE VARIANCE? Sigma2 and the mean square for error. WHAT OTHER MEASURES OF VARIABILITY CAN BE EASILY COMPUTED ONCE YOU KNOW THE VARIANCE? The standard deviation and the standard error of the sample mean. How do you compute THE VARIANCE? Find the distance of each score from the mean, square it, sum them up and divide by the number of scores in the population. THE STANDARD DEVIATION? Compute the square root of the variance. THE STANDARD ERROR OF THE SAMPLE MEAN? Divide the variance by n, the size of the sample, and then take a square root.