Standardized Score, probability & Normal Distribution Ibrahim Altubasi, PT, PhD The University of Jordan

Standardized Score Q: You got 76 in stats course. How did you do in the stats course? Which kind of information you will need for answering this question? Two distributions of exam scores. For both distributions, μ = 70, but for one distribution, σ = 3, and for the other σ = 12. The position of X = 76 is very different for these two distributions

Standardized Score If we know the population mean (μ) and the SD (σ) of a distribution and if we know a raw score X, we will know the position of this raw score in the distribution through the formula: (X – μ) / σ. This is a way to transform the raw data. The transformed score is z-score. Each z score will tell the exact location (how far away this score is from the mean) of the original X value within the distribution. z = (X – μ) / σ Z score is a standardized score with a sign that indicates direction from the mean (+ above the mean and - below the mean), and a numerical value equal to the distance from the mean measured in standard deviations.

Standardized Score We can transform an entire population of raw scores into z-scores. The shape of the distribution of the z-scores will be the same as that of the raw score. Q: What is the mean and SD for the transformed z-scores?

Standardized Scores After the raw scores are transformed into z-scores, the mean for the z-scores distribution is 0, and the standard deviation is 1

Standardized Scores Following a z-score transformation, the X-axis is relabeled in z-score units. The distance that is equivalent to 1 standard deviation on the X-axis (σ = 10 points in this example) corresponds to 1 point on the z-score scale.

Standardized Scores In addition to transform a raw score distribution into a standardized z-score distribution (where mean is 0 and SD is 1), we can also transform a raw score distribution into a standardized distribution with a predetermined mean and SD Examples of standardized distribution: SAT (μ=500, σ=100); IQ (μ=100, σ=15); T-score (μ=50, σ=10)

Standardized Scores A distribution with μ= 35 and σ= 8 is being standardized so that the new mean and standard deviation will be μ= 50 and σ= 10. 1) When the distribution is standardized, what is the value corresponding to a score of 39 from the original distribution? 2) What is the value in the original distribution corresponding to a score of 40 in the standard distribution?

Probability

Probability The role of probability in inferential statistics: Probability is used to predict what kind of samples are likely to be obtained from a population. Thus, probability establishes a connection between samples and populations. Inferential statistics rely on this connection when they use sample data as the basis for making conclusions about populations.

Probability Probability is defined as a proportion (or relative frequency), a specific part out of the whole set of possibilities. p(an event) = number of the outcomes classified as the event / total number of possible outcomes in the outcome space Tossing an unbiased coin: total possible outcomes (outcome space): head, tail The event: head, the probability that we get a head is 1/2 Selecting a card from a complete deck of cards: number of total possible outcomes: 52 The event: the king of hearts, p(the king of hearts) = 1/52 The event: a spade, p(spade) = 13/52 The event: a queen, p(queen) = 4/52. Selecting a marble from jar containing 8 black marbles and 7 white marbles: number of total possible outcomes: 15 The event: a black marble, p(a black marble) =

probability Addition Rule: If A and B are events, the probability of obtaining either of them is: P(A or B) = P(A) + P(B) - P(A and B) SS Space Mutually Exclusive Events do not share any outcomes in common A B Example: 1. Suppose the rehabilitation sciences faculty consists of 25% juniors, 15% seniors, and the remaining 60% is students of other grades. The relative frequency of students who are either juniors or seniors is 40%. We can add the relative frequencies of juniors and seniors because no student can be both junior and senior. P(Junior or Senior) = 0.25 + 0.15 = 0.40

Probability Addition Rule: If A and B are events, the probability of obtaining either of them is: P(A or B) = P(A) + P(B) - P(A and B) A B Space A and B Example: 2. Suppose that we draw one card from a deck of 52 playing cards. What is the probability that the card will be either a king or a heart? The probability of drawing a king is 4/52; the probability of drawing a heart is 13/52; and the probability of drawing the king of hearts is 1/52. Therefore: P(King or Heart) = 4/52 + 13/52 - 1/52 = 16/52=4/13

Probability Multiplicative Rule: If A and B are independent events, the probability of obtaining both of them is: P(A and B) = P(A) x P(B) Events A and B are said to be Independent when the occurrence of A is not dependent upon the occurrence of B and vice versa Example: Suppose we have two dice. A is the event that 4 shows on the first die, and B is the event that 4 shows on the second die. If both dice are rolled at once, what is the probability that two 4s occur? P(A) = 1/6 P(B) = 1/6 P(A and B) = P(A) x P(B) = 1/6 x 1/6 = 1/36

Probability Random sample: Each individual in the population has an equal chance of being selected. If more than one individual is to be selected for the sample, there must be constant probability for each and every selection. Selecting 2 cards from a complete deck of cards: For the 1st draw, p(Jack of diamonds) = 1/52; For the 2nd draw, p(Jack of diamonds) = ??. Sampling with replacement: A sampling technique that returns the current selection to the population before the next selection is made. A required part of random sampling A jar contains 10 red marbles and 30 blue marbles, if you take a random sample of n=3 marbles from the jar (sampling with replacement) and the first two marbles are both blue, what is the probability that the 3rd marble will be blue?

Probability A probability distribution is formed when a frequency distribution is converted into a relative frequency distribution. A frequency distribution histogram for a population that consists of N = 10 scores. If I randomly select an X from the population, what is the probability of getting Jim? How about an X=1? total possible outcomes: ??? the event: Jim, p(Jim)=1/10; the event: X=1, p(X=1) =f/N = ? What is the probability of getting an X=4? How about X>4?

Normal Distribution

Normal Distribution The Normal Distribution In simpler terms, the normal distribution is symmetrical with a single mode in the middle. The frequency tapers off as you move farther from the middle in either direction. The exact shape of the normal distribution is specified by an equation relating each X value (score) with each Y value (relative frequency). The equation is (π and e are mathematical constants.)

Normal Distribution A mathematical idealization of a particular type of symmetric distribution A mathematical curve that provides a good model of relative frequency distributions found in research The distribution of means calculated for many samples of subjects drawn from the same population can be approximated by the normal distribution when the sample size increases, even though the distribution of scores in the population is not normal (central limit theorem)

Normal Distribution Properties of Normal Distribution: Unimodal Symmetric about its mean The mean, mode, and median of the distribution are all equal Asymptotic: The curve never touches the X axis

Normal Distribution The Standard normal distribution The normal distribution following a z-score transformation.

Normal Distribution The population mean in a test is 200 and the SD is 50. Let’s assume that the population distribution of the test scores is normal. Then we can say that 68.26% of the population will have z-scores between -1 and +1 (corresponding to 150 and 250). 95.44% of the population will have z-scores between -2 and +2 (corresponding to 100 and 300).

Normal Distribution The population mean in a test is 200 and the SD is 50. Let’s assume that the population distribution of the test scores is normal. If we randomly select a student from this population, the probability that the student’s score falls within 150 to 250 is 68.26%. The probability that the student’s score is more than 150 will be 84.13%. The probability that the student’s score is greater than 300 is only 2.28%. [The probability that one gets a score 300 or higher is so small that we may conclude that this person is not selected from the population with mean 200 and SD 50]

Normal Distribution Adult heights form a normal distribution with a mean of 68 inches and a SD of 6 inches. If we randomly select one from this population, the probability that his height will be 5 feet 2 inches (62 inches) or shorter is …. How about the probability that his height will be 6 feet 5 inches (77 inches) or higher? How about 70 inches or higher?