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- Lesson Objectives At the end of this lesson, you should be able to Illustrates a Normal Random Variable and its Characteristics; and convert normal random variable to standard normal variable and vice versa.
- Normal Distribution NORMAL DISTRIBUTION • It is a continuous, bell-shaped curve distribution whose left and right tails extend indefinitely but come infinitely and never touch the x-axis. • The peak of the curve is the position of the mean which is found at the center of the distribution. The distribution thins out as the values move away from the mean
- Normal Distribution PROPERTIES OF NORMAL DISTRIBUTION • The distribution curve is bell shaped • The curve is symmetrical about its center • The mean, median and mode coincide at the corner • The tails flatten out indefinitely along the horizontal axis but never touch it ( the curve is asymptotic to the base line) • The are under the curve is 1 or 100%
- Normal Distribution 2 FACTORS THAT THE GRAPH OF THE NORMAL DISTRIBUTION MAY DEPEND IN 1. MEAN • The change of values of the mean shifts the graph of the normal curve to the right or to the left
- Normal Distribution 2 FACTORS THAT THE GRAPH OF THE NORMAL DISTRIBUTION MAY DEPEND IN 1. Standard deviation • Large value of standard makes the normal curve short and wide • Small value of standard deviation yield a skinnier and taller normal curve
- Normal Distribution Empirical Rule • Is also referred to the 68% 95%, and 99.7% • It tells us that for normally distributed variable the following are true;
- Normal Distribution Empirical Rule • Is also referred to the 68% 95%, and 99.7% • It tells us that for normally distributed variable the following are true; • Approximately 68% of the data lie within 1 standard deviation of the mean • Approximately 95% of the data lie within 2 standard deviation of the mean • Approximately 99.7% of the data lie within 3 standard deviation of the mean
- Normal Distribution EXAMPLE 1: Scores in examination the scores of the senior high school students in Colegio De Sto Nino in their Statistics and Probability Examination are normally distributed with a mean 35 and the standard deviation of 5 A. What percent of the score are between 30-40? B. What scores fall within 95% of the distribution
- Normal Distribution EXAMPLE 1: Scores in examination the scores of the senior high school students in Colegio De Sto Nino in their Statistics and Probability Examination are normally distributed with a mean 35 and the standard deviation of 5 A. What percent of the score are between 30-40? B. What scores fall within 95% of the distribution
- Normal Distribution STANDARD NORMAL DISTRIBUTION 1. The total area under the standard normal curve is equal to 1. 2. The standard normal distribution is symmetrical about the mean. This means that the area of the left and right side of the mean is 0.5. 3. The mean, median and mode are equal. 4. The tails of a standard normal curve are asymptotic relative to the horizontal line. 5. The probability that the standard random variable Z will fall into an interval a to b is equal to the area of the shaded region under the standard normal curve between two points
- Normal Distribution Understanding the Standard Normal Curve The standard normal curve is a normal probability distribution that is most commonly used as a model in inferential statistics. The equation that describes a normal curve is: 𝛾 = 𝑒 − 1 2 ( 𝑋−𝜇 𝜎 )2 𝜎 2𝜋 Where: Ƴ = height of the curve particular values of X X = any score in the distribution σ = standard deviation of the population µ = mean of the population π = 3.1416 e = 2.7183
- Normal Distribution Understanding the Standard Normal Curve • By substituting the mean, µ = 0 and the standard deviation, σ =1 in the formula, mathematicians are able to find areas under the normal curve. • The area between -3 and +3 is almost 100% because the curve almost touches the horizontal line. Thus, there is a small fraction of the area at the tails of the distribution.
- Normal Distribution Table of Areas under the Normal Curve • The z-score is a measure of relative standing. It is calculated by subtracting sample/population mean form the measurement X and then dividing the result by the corresponding standard deviation. The final result, the z-score, represent the distance between a given measurement X and the mean, expressed in standard deviations. Either the z-score locates X within a sample or within a population 𝑧 = 𝑋−𝜇 𝜎 (population data) 𝑧 = 𝑋−X̄ 𝑠 (sample data) Raw scores may be composed of large values, but large values cannot be accommodated at the base line of the normal curve. So, they have to be transformed into scores for convenience without sacrificing meanings associated with the raw scores.
- Normal Distribution Importance of using z-scores? • Raw scores may be composed of large values, but large values cannot be accommodated at the base line of the normal curve. So, they have to be transformed into scores for convenience without sacrificing meanings associated with the raw scores. • The z-values are matched with specific areas under the normal curve in a normal distribution table. Therefore, if we wish to find the percentage associated with X, we must find its matches z-value using the z-formula.
- Normal Distribution EXAMPLE 1: Reading Scores Given the mean, µ = 50 and the standard deviation σ = 4 of a population of reading scores. Find the z-value that corresponds to a score X=58. 𝑧 = 𝑋−𝜇 𝜎 𝑧 = 58−50 4 𝑧 = 8 4 = 2.00 Thus, the z-value that corresponds to the raw score 58 is 2 in a probability distribution
- Normal Distribution EXAMPLE 1: Reading Scores As you can see, score X = 58 corresponds to z=2.00. It is above the mean. So, we can say that, with respect to the mean, the score of 58 is above average.
- Normal Distribution EXAMPLE 2: Scores in Science Test Given X = 20, X̄ = 26 and s = 4. Compute the corresponding z-score. 𝑧 = 𝑋−X̄ 𝑠 𝑧 = 20−26 4 𝑧 = −6 4 = −3 2 = -1.50 The corresponding z-score is -1.5 to the left of the mean/the z-value that corresponds to 20 is -1.50 in a sample distribution. With respect to the sample mean, the score 20 is below the sample mean
- Normal Distribution EXAMPLE 3: Suppose the corresponding z-score of the weight of employees in Meycauayan City Hall is 3.5, what should be the weight of an employee if it resembles a normal distribution with a mean 58 kg and a standard deviation of 2.
- Normal Distribution Let’s Try! A. State whether the z-score locates the raw score X within a sample or within a population 1. X = 50, s = 5, X̄ = 40 2. X = 40, σ = 8, 𝜇 = 52 3. X = 36, s = 6, X̄ = 28 4. X = 74, s = 10, X̄ =60 5. X = 82, σ = 15, 𝜇 =75 B. From exercise A, state whether each raw score lies below or above the mean.
- Normal Distribution Z-TABLE
- Normal Distribution Z-TABLE
- Normal Distribution Four-step process in Finding the Areas Under the Normal Curve Given a z-Value 1. Express the given z-value into a three-digit form. 2. Using the z-table, find the first two digits on the left column. 3. Match the third digit with the appropriate column on the right 4. Read the area (or probability) at the intersection of the row and the column. This is the required area.
- Normal Distribution EXAMPLES • Find the area that corresponds to z=1 (Finding the area that corresponds to is the same as finding the area between z=0 and z=1) • Find the area that corresponds to z=1.36 • Find the area that corresponds to z= - 2.58 (The area that corresponds to z=2.58 is the same as the area that corresponds to z= - 2.58)
- Normal Distribution Let’s Try! Find the corresponding area between z = 0 and each of the following: 1. z = 0.96 2. z = - 1.74 3. z = 2.18 4. z = - 2.69 5. z = 3.00

- Z-score / standard score – indicates how many standard deviations an element is from the mean

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