Chap 4 Frequency Probability and Percentage Graphs Normal

Chap 4 - Frequency, Probability and Percentage Graphs - Normal Distribution - z scores

Frequency, Probability and Percentage Graphs • Draw a frequency graph for the following distribution 2, 6, 6, 11, 11, 18, 18, 29, 29, 37, 43 X 2 6 11 18 29 37 43 f 1 2 3 4 3 2 1

• Draw a probability graph for the following distribution 2, 6, 6, 11, 11, 18, 18, 29, 29, 37, 43 X 2 6 11 18 29 37 43 f 1 2 3 4 3 2 1 Probability 1/16 = 0. 0625 2/16 = 0. 125 3/16 = 0. 1875 4/16 = 0. 25 3/16 = 0. 1875 2/16 = 0. 125 1/16 = 0. 0625

• Draw a percentage graph for the following distribution 2, 6, 6, 11, 11, 18, 18, 29, 29, 37, 43 X 2 6 11 18 29 37 43 f 1 2 3 4 3 2 1 Probability 1/16 = 0. 0625 2/16 = 0. 125 3/16 = 0. 1875 4/16 = 0. 25 3/16 = 0. 1875 2/16 = 0. 125 1/16 = 0. 0625 Percentage 0. 0625 x 100= 6. 25 0. 125 x 100 = 12. 5

Based on your probability graph • • What is the probability of finding scores a) more than 18 in this distribution? b) less than 6 in this distribution? c) more than 37 in this distribution?

Normal Distribution • Is the distribution of scores that has the following characteristics: • 1)Is unimodal, meaning has one hump • 2) symmetic – the right half of the curve is a mirror image of the left half • 3) has the same value of mean, median and mode • 4) is asymptotic – the tails of the distribution never touch the abscissa or the x-axis • 5) has 68% of the scores of the distribution lie between µ ± σ. • 6) has the following mathematical model

Normal Distribution Frequency / Probability / Percentage 13. 59% 34. 13% 13. 59% Scores 1σ 1σ Mean Median Mod 50% of the scores are found on each side of the mean 68. 26% of the scores are found between 1 standard deviation from the mean

z scores • A z score is a measure of how many standard deviations a raw score is from the mean of the distribution • Suppose a normal distribution has a mean of 20 and a standard deviation (SD) of 4, the score 24 is one SD above the mean. As such the score 24 would be a 1 z score above the mean (z = +1) • What z score will be assigned to a score of 16? (Ans: z = -1)

A z score is also called a standard unit or standard score When all the raw scores of a normal distribution have been transformed to z scores, the resulting distribution is called standard normal distribution. The standard normal distribution has a mean of 0 and a SD of 1. Standard Normal Distribution Frequency / Probability / Percentage 13. 59% 2 z 1 z SD = 1 34. 13% 13. 59% 1 z SD = 1 Mean = 0 Median Mod SD = 1 2 z z scores

How to convert a raw score to a z score? Population Sample Z=X-µ σ Z = X - ___ s Where X = the raw score to be transformed = the mean of the sample s = the sample SD µ = the mean of the population σ = the population SD

Example • For a population distribution with µ = 4. 80 and σ = 2. 14, what is the z score of a raw score of 6? Z=X-µ σ Z = 6 – 4. 8 = +. 56 2. 14 The raw score of 6 has a standard score z = +. 56 Which means that the score 6 is. 56 standard deviations above (+) the mean

Characteristics of z scores • 1) A z-score distribution is obtained by transforming every raw score to a z score. • 2) A z-score distribution always has a mean of 0 and a standard deviation of 1. • 3) All raw score that fall below the mean have some z value that is negative; all raw scores that fall above the mean has positive z scores. • 4) A raw score that is one SD above the mean has a z score of +1 and a raw score that is one SD below the mean has a z score of -1. • 5) A raw score that is the same as the mean has a z value of 0 • 6) The z –score distribution will have the same shape as the raw score distribution

Convert the following raw scores to z scores given mean = 8 and SD = 4 • • • a) 12 b) 8 c) 4 d) 16 e) 6

Area under the curve Area is a measure of the probability of finding a score Standard Normal Distribution Frequency / Probability / Percentage 13. 59% -2 z 34. 13% -1 z SD = 1 34. 13% 13. 59% 1 z SD = 1 2 z z scores Mean = 0 Median Mod 68. 26% of the scores fall between -1 z and +1 z. This means the probability of a randomly selected score to fall between - 1 z and +1 z is. 6813

Area under the curve Standard Normal Distribution Frequency / Probability / Percentage 2. 28 13. 59% -2 z 34. 13% -1 z SD = 1 34. 13% 13. 59% 1 z SD = 1 2. 28 2 z z scores Mean z = 0 Median Mod 50% of the z scores exceed the Mean = 0. This means the probability of a randomly selected score to be found above the Mean is. 50

Area under the curve z Area between Mean and z Area Beyond z 0 0 +1 . 3413 . 5 -. 3413 =. 1587 +2 . 4772 . 5 -. 4772 =. 0228 +1. 5 . 50 (see table for normal distribution – get the area beyond z) =. 0668 The area beyond the z score is given by the Normal Distribution Table Open your Table book and learn how to use it how to use the Normal Distribution Table

Exercise 1 • You are given a normal distribution with the mean of 12 and SD of 4. Using this distribution, calculate the percentage and probability of scores that • a) fall between 8 and 16 • b) exceed 18 • c) fall below 6 Draw a normal distribution graph to show these positions

What percentage of scores fall between the z scores of +. 25 and + 1. 20? Frequency / Probability / Percentage +. 25 Mean z = 0 Median Mod +1. 20 z scores

Exercise 2: Draw the graph to show the area involved • What percentage of scores fall between the z scores of – a) -. 25 and - 1. 20 • b) -. 20 and +. 20 • c) below the z score of -1. 96 • d) above the z score of +1. 96 • e) below the z score of -1. 96 and above the z score of +1. 96 • f) -1. 2 and +2. 5 • g) below the z score of -. 4 and above the z score of -. 6

Exercise 3 : Draw the graph to show the area involved • Given the Mean of a population is 100 and the SD = 25, what percentage of scores fall between the scores • a) 75 and 125 • b) above 125 • c) below 75 • d) above 80 • e) above 110 • f) between 80 and 100 • g) between 60 and 130 • h) below 60 and above 130

Homework • You obtained the following data on students’ self concept for your Masters Project Paper Research. 23, 42, 45, 25, 43, 12, 23, 32, 33, 30, 32, 26, 33, 47, 55, 40, 28, 23, 16, 47, 33, 35, 50, 48, 8, 30, 23, 35, 29, 36, 33, 18, 26, 37. 33, 15, 33 a) Describe the data using measures of central tendency: Mean, median and mode b) Describe the data using measures of variability: Range, Mean deviation, Variance and Standard Deviation. c) Assuming the distribution is normal, use the normal distribution table and answer the following questions: i) what percentage of scores exceed the score 35. ii) what percentage of scores lie between 30 and 46. iii) what percentage of scores lie between 20 and 50 iv) what is the probability of obtaining a score greater than 50? v) what is the probability of obtaining a score greater that 53? d) Draw graphs for each question in c) to show the area involved. e) Present your descriptive statistical findings in a table (use the APA format).