Lesson UNDERSTANDING THE Z- 4 SCORES Learning Concept Let us begin understanding the z-scores by acknowledging that for a given distribution, it is more preferred to have a larger set of data in order to make good generalizations. However at times in a given distribution, raw scores may be composed of large values and large values cannot be accommodated at the baseline of the normal curve. Thus, the raw scores have to be transformed into z-scores in order to get meaningful decisions relative to the concepts of finding the equivalent percentage and probability of the given measure of value from the mean. For any population, the mean and the standard deviation are fixed. This gives the way to understand that for every raw score X, there corresponds an exactly one z- score value, and vice versa. Therefore, if we wish to find the percentage associated with X, we must find its matched z-value using the z formula. The z formula is given by: Z= X- A (z-score for population data) _X- X (z-score for sample data) where X = the raw score A = population mean X = sample mean = population standard deviation s = sample standard deviation 25The area of the region under the normal curve represents the probability or percentage or proportion of a given measurement value. It is computed by subtracting the measurement X to the sample X or population mean, then dividing the result by the standard deviation o. The resulting value gives the z-score. The z-score indicates the distance between a given measurement X and the mean expressed in standard deviations. It locates either within a sample or within a population. But for now, a readily available z-Table is made to obtain the corresponding area given a z-score. Steps in finding the z-score given the mean (X or /) . standard deviation (@) and the measurement (X) 1. Use the formula z= _"for finding the z-score 2. Check the given values: mean (Xor /) . standard deviation () and the measurement (X) 3. Substitute the given values in the formula 4. Compute for the z-value Illustration. 1. In Mrs. Shirley's science class, a student gained a score of 46. What is the z- value of his score if test result has population mean 45 with standard deviation of 2? Steps Solution a. Use the formula for finding the z- scores of population data b. Check the given values H = 45, 0 = 2, X = 46 c. Substitute the given values in the 46-45 2 formula d. Compute the z-value Z= = =0.5 Therefore, the z-value that corresponds to the raw score of 46 is 0.5. 260.5 39 43 45 46 47 51 Mean 2. Leslie got a score of 68 in an examination she participated. What is the corresponding z-value of her score if the result has # = 75 and o = 5. Steps Solution a. Use the formula for finding the z- z= = scores of population data b. Check the given values H = 75, 0 = 5, X = 68 c. Substitute the given values in the Z= 58 - 75 formula d. Compute the z-value 2= = - 1.4 Therefore, the z-value that corresponds to the raw score of 68 is -1.4. 273. Mrs. Bastillada conducted a test survey about determining the current level of students' learning on Geometry. After checking the test questionnaires, she was amazed that a student attained 50 correct answers. What is the corresponding z-value of the student's score if the sample mean of the test results was 45 and standard deviation is 6? Steps Solution a. Use the formula for finding the z- scores of sample data b. Check the given values X = 45, a = 6, X = 50 c. Substitute the given values in the 2 50-45 formula d. Compute the z-value z= = = 0.83 Then 0.83 e z-value that corresponds to the rawscore of 50 is 0.83. 27 33 39 50 51 57 63 28What I Can Do Exercise 1 You are applying for a job that requires a score on either a P Test or an A Test. The company needs only one test result. You have acquired scores on both tests. You've scored 98 on the P Test and 112 on the A Test. Which test results will you most likely to submit? Is the information enough for you to convince yourself on submitting your chosen test result? Explain why. Exercise 2 Given A = 110 and o = 10 in a college entrance examination, find the corresponding z-value of the scores gained by the following students. Show all your solutions and sketch its graph. 1. X = 98 2. X = 65 3. X = 120 4. X = 74 5. X = 135