Normal Distribution Curve PDF

NDC Normal Distribution Curve Also called: Bell Curve or Gaussian Curve (named after Karl Friedrich Gauss)  Normal Distributions possess the following: 1. Normal distributions are symmetric around their mean. 2...

NDC Normal Distribution Curve Also called: Bell Curve or Gaussian Curve (named after Karl Friedrich Gauss)  Normal Distributions possess the following: 1. Normal distributions are symmetric around their mean. 2. The mean, median, and mode are equal. 3. Normal distributions are denser in the center and less dense in the tails (data values tend to be toward the center of the baseline). 4. The baseline (x-axis) is divided into 6-sigma distances. 5. The tails of the distribution and the x-axis do not touch (i.e., asymptotic).  Normal Distributions possess the following: 6. The area under the normal curve is equal to 1.0. 7. Normal distributions are defined by two parameters, the mean (μ) and the standard deviation (σ). 8. 68% of the area of a normal distribution is within one standard deviation of the mean. 9. Approximately 95% of the area of a normal distribution is within two standard deviations of the mean. 9. Approximately 99.7% of the area of a normal distribution is within three SDs of the mean. Class Boundaries Percentages of the normal distribution  Try this for your understanding: 1. For example, 68% of the distribution is within one standard deviation (above and below) of the mean, and approximately 95% of the distribution is within two standard deviations of the mean. Therefore, if you had a normal distribution with a mean of 50 and a standard deviation of 10, then 68% of the distribution would be between 50 – 10 = 40 and 50 +10 =60. Similarly, about 95% of the distribution would be between 50 - 2 x 10 = 30 and 50 + 2 x 10 = 70. 1. If we take into consideration all values (i.e., the total area under the curve), it will be equal to 100% or, in other words, 1.0. If we are, for instance, interested in a score of 90, this means we look at the from 85.5 to 94.5 This means that the proportion of the total area of the curve where we can find a score of 90 is equal to 0.1359. Because the total area is equal to 1.0, we can also easily subtract 0.1359 from this and conclude that the proportion of the area outside is equal to 0.8641. which is 13.59% of the distribution curve. As the standard deviation gets smaller, the distribution becomes much narrower, regardless of where the center of the distribution (mean) is. 1. Exam Scores Distribution 1. Problem: A teacher conducted a math exam for a class of 30 students. The average (mean) score was 75, and the standard deviation was 5. The exam scores followed a normal distribution. 2. Question: What percentage of students scored between 70 and 80? 3. Answer: Approximately 68% (within one SD of the mean).  2. Product Weights in a Factory 1. Problem: A factory produces bags of rice, where the weights are normally distributed. The mean weight is 50 kg, and the standard deviation is 2 kg. 2. Question: What percentage of bags weigh less than 47 kg? 3. Answer: 9.08% 1. 3. Situation: A teacher wants to analyze the test scores of a recent statistics exam for a class of 50 students. The scores are normally distributed with a mean (μ) of 75 and a standard deviation (σ) of 10. 2. Problem: 3. 1. What percentage of students scored between 65 and 85? 4. 2. If a student scored 95, how many standard deviations above the mean is their score? 5. 3. What is the probability that a randomly selected student scored less than 65?  Measures of Shape or Measures of Tails 1. A. Skewness describes how symmetrical or asymmetrical the data is, and it indicates the position of the tail. 2. It measures the asymmetry of the data distributions and helps us understand how values are spread around the mean. 3. Generally, there are two tails in a distribution, located at the far left and far right of the distribution where the curve visually ends. 1. No Skew 1. In an NDC, the tails are symmetrical, and there is no skewness present. When data is not skewed, we then know that the bulk of the data values are located near the center of the baseline and spread towards the sides from there. In the example above, the mean is 78 and is located at the center of the baseline. And because the curve is not skewed, we also know that the value 78 also represents the median and the mode. 2. Positively Skewed Distribution 1. A positively skewed distribution has its longer tail located at the right side of the distribution, where the positive side of the baseline is. 2. Positively Skewed Distribution 1. A positively skewed distribution (i.e., right-tailed) shows that the bulk of the data is located at the left side of the baseline or the negative side. In assessment, this indicates that most scores lean towards lower values. As such, we may assume that either the test is very difficult or the ability of test- takers may be somewhat impaired (possibly due to environment or personal factors). In the example, we can see that most scores are within the range 54 to 70. 3. Negatively Skewed Distribution 1. A negatively skewed distribution has a tail at the left side, where the negative side of the baseline is. 3. Negatively Skewed Distribution 1. In a negatively skewed distribution (i.e., left-tailed), most of the scores would be located on the right side or the positive side of the baseline. This may mean that the test is very easy , which leads to most test-takers achieving high scores. Alternatively, there may also be errors in test development and/or administration, or the test- takers may also possess above-average mental abilities. SUMMARY: Measures of Shape or Measures of Tail B. Kurtosis or the weight of the tails: measures the “tailedness” of a distribution 1. 1. Leptokurtic: It is a curve having a higher peak than a normal curve, has low tails with a peak easily distinguishable; data are too concentrated near the center. 2. 2. Platykurtic: does not have a visible "peak" or tails and is rather flat; there is less concentration of scores near the center. 1. 3. Mesokurtic: It is a curve having a normal peak or normal curve. There is equal distribution around the center value (Mean). It fits the NDC. Summary: