PY 427 Statistics 1Fall 2006 Kin Ching Kong, Ph.D Lecture 3 Chicago School of Professional Psychology

Agenda Variability Definition The Range Population Standard Deviation and Variance Sample Standard Deviation and Variance z-Scores and Standardized Distributions Intro & Definition Purpose of z-Scores z-Score Formula Standardized Distributions Looking Ahead to Inferential Statistics

Variability Definition: variability in statistics is a quantitative measure of the degree to which scores in a distribution are spread out (large variability) or clustered together (little variability). Variability tells us: How much distance is expected between scores, and between a score and a mean. How representative is a score or a set of score. Or, how large a sampling error to expect.

The Range The Range: the distance between the largest score and the smallest score in the distribution. Range = URL X max – LRL X min The interquartile range: the range covered by the middle 50% of the distribution Interquartile range = Q3 – Q1

Standard Deviation, Conceptual Standard Deviation: the average distance from the mean. Standard Deviation: Conceptual 1. Deviation: the distance and direction from the mean: Deviation score = x –  2.  deviation scores) = 0 3. Square each deviation scores 4. mean squared deviation = variance 5. Standard Deviation = Square root (variance)

Standard Deviation, Definitional Formula Standard Deviation for a Population, Definition: 1. Deviation score = x –  2. (x –  ) 2 3.  (x –  ) 2 = SS = Sum of Squared Deviations 4.  (x –  ) 2 /N = SS/N = MS = Mean Squared Deviation = Variance,  2 5. Standard Deviation,  = Square root (variance) = Square Root (SS/N)

Standard Deviation, Computational Formula Standard Deviation for a Population, Computational Formula 1. SS =  X 2 – (  X) 2 /N 2. Variance =  2 = SS/N 3. Standard Deviation =  = Square root (variance) = Square Root (SS/N)

Standard Deviation, Exercise Find the Variance and Standard Deviation for the following populations of scores: a) 7, 1, 7, 9 (use the definitional formula) b) 1, 6, 1, 1, 1 (use the computational formula) Answers

Standard Deviation, for a Sample Standard Deviation for a Sample Sample variability underestimates Population variability. Figure 4.6 of your bookFigure 4.6 of your book 1. SS =  (X – M) 2 Computational Formula: SS =  X 2 – (  X) 2 /n 2. Variance = s 2 = SS/(n-1) 3. Standard Deviation = s = Square root (variance) = Square Root (SS/(n-1))

Standard Deviation, Exercise Find the Variance and Standard Deviation for the following samples: a)5, 1, 5, 5 (use definitional formula) b)1, 7, 1, 1 (use computational formula) Answers

Degree of Freedom Degrees of Freedom = the number of scores free to vary. Degrees of Freedom, df, for sample variance: df = n – 1 S2 = SS/df = SS/(n-1)

Biased & Unbiased Statistics A sample statistic is an unbiased statistic if the average value of the sample statistic, obtained over many different samples, is equal to the population parameter. E.g. Table 4.1 of your book

z-Scores, Introduction A college student got a score of 76 on a biology exam. Did the student do well? Figure 5.1 of your book

z-Scores, Definition A z-score specifies the precise location of each X value within a distribution. The sign of the z-score signifies whether the score is above or below the mean. The numerical value specifies the distance from the mean in terms of the standard deviation. e.g. z= +2 means the raw score is located 2 standard deviation above the mean.

z-Scores, Purposes Two purposes: Each z-score identifies the exact location of the raw score within the distribution. The z-scores form a standardized distribution that can be directly compared to other distributions that also have been transformed into z-scores

z-Scores, Formula The z-Score Formula: z = (X –  )/  Exercise: 1. A distribution has a  = 50,  = 5, what z- score correspond to a raw score of 60? 2. A distribution has a  = 86,  = 7, what is the z-score for the raw score of X = For a distribution with a  = 100,  = 50, what X value correspond to z = -1.5? Answers

Standardized Distributions A standardized distribution is composed of scores that have been transformed to create predetermined values for  and . Standardized distributions are used to make dissimilar distributions comparable. A distribution of z-scores is a standardized distribution with a  of zero and a  of one. The shape of the z-score distribution is the same as the original raw score distribution. Figure 5.3 in your book

Standardized Distributions Other Standardized Distributions: z-scores can be used to create other standardized distributions with any pre-determined  and  (e.g. SAT, IQ) Step 1: The original raw scores are transformed into z-scores Step 2: The z-scores are then transformed into new X values based on the predetermined  and .

Standardized Distributions, Example E.g. 5.6 of your book: The distribution of raw scores on an exam has a  of 57 and  of 14. The instructor wants to simplify the distribution by transforming all the scores into a new, standardized distribution with  = 50 and  = 10. Mary got a raw score of X = 64 Joe got a raw score of X = 43 What are their standardized scores? Figure 5.5 of your book

Looking Ahead to Inferential Statistics Figure 5.6 of your book Example 5.7 The distribution of adult rat weights is normal with a  of 400g, and a  of 20g. A research selects one newborn rat and injects it with a growth hormone. When this rat reaches maturity, it’s weight is obtained to determine if the hormone had an effect. Figure 5.7 of your book