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PSY_1004_2023H1_session 05.pdf
- 1. PSYC1004 Introduction to quantitative methods in psychology Session 5 1
- 2. Number Mean Deviation Squared deviations 1 3.20 -2.2 4.84 3 3.20 -0.2 0.04 2 3.20 -1.2 1.44 4 3.20 0.8 0.64 6 3.20 2.8 7.84 1 3.20 -2.2 4.84 1 3.20 -2.2 4.84 3 3.20 -0.2 0.04 7 3.20 3.8 14.44 4 3.20 0.8 0.64 39.6 4.40 2.10 Standard deviation (example) 2 Sample variance and SD ( i.e., 39.6/9) (SQRT of 4.40) (sum of squared deviations) square root of 4.40
- 3. The z score of a data point X M = the mean of the dataset S = the SD of the dataset z score 3 Math (80 - 82) / 6 = -0.33 Verbal (75 - 75) / 3 = 0.00 Science (70 - 60) / 5 = 2.00 Logic (77 - 70) / 7 = 1.00 Z score (Caldwell) The z score of a raw score in a data set is the distance of the data point from the mean in standard deviation units eg both above average > use sd and z score to see which perform better sd cant be negative Text Text
- 4. z score It can be used to locate a score in a distribution of data: It informs whether the score is above or below the mean; and the scores deviation from the mean is relatively large or relatively small compared with the typical deviations in the dataset. 4 Example: If a given z score is negative, the raw score being represented is (above/ below/ at) the population mean? Which of the following z scores represents a raw score that is the most atypical (i.e., farthest from the mean)? (a) 3.10 (b) -0.82 (c) 0.47 (d) 2.20 below a
- 5. Histogram 5 ... When scores are measured on a continuous variable, the distribution of frequency (number of cases or units in each value range) is commonly displayed by a histogram. z core will be the same as the mean and the sd are the same but the percentile not the same percentile and z score is depend on the distribution
- 6. Population distribution The distribution of a continuous variable in a large/infinite population is typically represented graphically using a continuous line (probability distribution curve). The histogram for a sample of data from the population can be taken as an approximation of the curve. 6 (Source: Field) pobility density the meaning of the curve larger sample > better approximation > line join become more like smooth cure
- 7. Probability distribution The probability that a continuous variable is between two specified values is equal to the area under the distribution curve over that interval. 7 Source: Howell Probability or relative frequency probability finding someone under the value = the area under the curve
- 8. Normal distribution It is a family of theoretical probability distributions precisely generated by a formula The distributions of many variables are taken/assumed as (or being close to) normal distributions A normal distribution is symmetrical about its mean and extends to infinity and negative infinity A bell-shaped probability distribution curve does not necessarily represent a normal distribution. 8 probability mean and sd > curve all normal distribution are bell shaped but bell shaped not eual to normal distribution
- 9. Normal distribution 9 The approximate areas under the standard normal distribution curve that lie between z = 0 and several whole-number z scores (Source: Hatcher) The percentage of the area under a normal distribution curve is a function of z score.
- 10. Normal distribution 10 Example: The probability of a standardized and normally distributed variable being less than 0.5, P (z < 0.5) = .6915 .6915 P( - < z < .5)
- 11. Normal distribution P (z < 0.84) = 0.8 As the total area under a standard normal distribution curve = 1, P (z > 0.84) = 1 0.8 = 0.2 As a normal distribution curve is symmetrical, P (z < - 0.84) = 0.2 11 0.8 .84 0.2 -.84 0.2
- 12. Example What is the probability that a standardized and normally distributed variable is between the mean (z = 0) and 1.48? 12 1.48 From the standard normal distribution table, probability (z < 1.48) = .9306; i.e., referring to the diagram, the area of A + B = 0.9306 Area of A = 0.5 (since a normal distribution is symmetrical about its mean) The required probability = the area of B = 0.9306 0.5 = 0.4306 B A
- 13. Normal distribution 13 P(z < .8) = .7881 The probability of z being between -1 and 0.8 P(z < -1) = . 1587 = .7881 - .1587 =.6294
- 14. Example The mean height of the individuals in a population is 175 cm and the standard deviation (SD) of their height values is 10 cm. Assuming that height is normally distributed in the population and an individual is randomly selected, what is the probability that the selected individuals height is between 165 and 185 cm? In terms of z scores, the range of 165 to 185 cm corresponds to (165 175)/10 and (185 175)/10, i.e., z = -1 to 1. The probability required, as obtained from the statistical table, is P (z < 1) P (z <-1) ~ 68.3% (P stands for probability) 14 only mean and sd (x-m)/sd > z score find the probability between the two z score z score can be apply no matter the distribution is normal or not when the normal distribution is used, the z score can be used to l link the z score and percentile