Normal Distribution PDF

NOTE: THIS IS A SIMPLIFIED VERSION OF THE LESSON: OPEN THE LEARNING CONTENT IN HURTADO FOR MORE IN DEPTH EXPLANATIONS AND EXAMPLES. (please see final note sa last page ) NORMAL DISTRIBUTION INFOGRAPHICS What is Normal Distribution? also known...

NOTE: THIS IS A SIMPLIFIED VERSION OF THE LESSON: OPEN THE LEARNING CONTENT IN HURTADO FOR MORE IN DEPTH EXPLANATIONS AND EXAMPLES. (please see final note sa last page ) NORMAL DISTRIBUTION INFOGRAPHICS What is Normal Distribution? also known as the Gaussian distribution, is It describes how the a continuous probability values of a variable are distribution that is distributed symmetrical and bell- shaped. According to the Central Limit Theorem, When you take many Also called normal curve, it is random samples of a large distribution of data where the enough size from any mean, median, and mode are population, the average of equal, the distribution is those samples will form a clustered at the center, the normal (bell-shaped) graph is a bell-shaped curve, distribution, even if the original and symmetrical. population's distribution is not normal. Properties 01 02 03 BELL SYMMETRY AVERAGES SHAPED The distribution is bell The line at the center of Mean, median and shaped. not square, the curve indicates mode coincide at the not rectangle, but symmetry. It divides the center. It means all of BELL. curve into 2 equal parts. them are located at the center 04 DISPERSION The width of the curve 05 ASYMPTOTIC The ends of the curve 06 AREA 1 The area under the is determined by the stretch out along the curve is 1. This standard deviation of horizontal axis, getting represents the the distribution. closer but never touching it. totality of data used This means the curve is on the curve. asymptotic to the baseline. USING THE NORMAL CURVE 01 STANDARD NORMAL CURVE A standard normal curve is a probability distribution that has a mean 𝜇 = 0 and a standard deviation 𝜎 = 1. We use standard normal curves with a mean of 0 and a standard deviation of 1 because they simplify calculations, allow easy comparison between datasets, help normalize data, and are foundational in many statistical methods. 02 -used when data is individual or measurement (x) is given Z-SCORE or VALUE A z-score tells you how far a data point is from the average, measured in standard deviations. It helps compare different data points, find probabilities, and identify outliers. - used when sample mean / We convert data to z-scores to standardize it, making sample size X is given. comparisons easier, calculating probabilities, and identifying outliers.  FINDING THE AREA OF A Z-SCORE Before finding the area, make sure that your z-score or value has been rounded to two decimal places ALWAYS DRAW THE CURVE FIRST. 01 Example: Find the area under the standard normal curve between 𝑧 = 0 and 𝑧 = 0.96. Finding the Area from 0 to z Look at your z-score. Find the first two digits in the first column, then find the last digit in the first row. The intersection of these values is the area. 0.96 First 2 digits 0.9 Last digit 0.06 02 Example: Find the area under the standard normal curve from 𝑧 = 1 and 𝑧 = 2 Between 2 z-values Find the area of each of the z-scores separately. The curve will tell you if you need to add or subtract their areas. Clue: if both z-scores have the same signs( both positive or both negative) you subtract their areas. Otherwise, add them. z = 1 -> A= 0.3413 0.4772 – 0.3413 = 0.1359 z = 2 -> A= 0.4772 03 Example: Find the area under the standard normal curve to the left of z = -1.5 Z-score to a direction Find the area of the z-score. Note the direction (to the left, right, less than, or greater than). Then, decide whether to add or subtract 0.5 Clue: The normal curve will indicate whether to add or subtract 0.5. A helpful observation is that the curve has two halves, the left and right sides. If you draw the curve and shade part of both halves, you add 0.5. If you only shade one half of the curve, you subtract 0.5. z = 1.5 -> A= 0.4332 0.5 – 0.4332 = 0.0668 PERCENTILE AND Z-SCORES 01 Percentile of a Z-score 1. Find the area (or probability) corresponding to the given z-score using a z-table or calculator. 2. Note the sign of the z-score. 3. If the z-score is positive, add 0.5 to the area. If it's negative, subtract the area from 0.5. 4. Simplify your answer to two decimal places. 5. Express it as a percentile rank. Example: Find the area under the standard normal curve to the left of z = -1.5 𝑧-score 2.34: A = 0.4904 Adding 0.5000 and round off to the nearest hundredths: 0.4904 + 0.5000 = 0.99 The 𝑧-score represents 99th percentile or P99 The shaded region is 99% of the distribution. 02 Z-score of a percentile 1. Convert your Percentile rank to a decimal 2. ALWAYS subtract 0.5 to that decimal 3. find the decimal (area) in the table, look for the exact value, if wala, look for the nearest one. 4. identify the z-score of that area. 5. if the percentile rank is less than 50, the z-score is negative, if its greater, its positive. P34 or 34th percentile (34%) represents the 0.3400 area under the normal curve. A = 0.5000 – 0.3400 = 0.1600 Nearest value = 0.1591 𝑧 = - 0.41  NOTES INFO 01 DISCRETE AND CONTINUOUS RANDOM VARIABLES Discrete random variables have countable outcomes with a limit, while continuous random variables have outcomes on a continuous scale, like height or temperature, with no limit. CLUE: Discrete random variables have outcomes that are whole numbers, while continuous random variables can have outcomes that include decimals. 02 DISCRETE PROBABILITY DISTRIBUTION AND HISTOGRAM A discrete probability distribution shows the values a random variable can take and their corresponding probabilities. A histogram is a graphical method used to display the shape of a distribution, especially useful for large datasets. 03 PROPERTIES OF A PROBABILITY DISTRIBUTION A probability distribution has two key properties: each probability must be between 0 and 1, and the sum of all probabilities must equal 1. 04 NOTES ON VARIANCE AND STANDARD DEVIATION Variance and standard deviation measure the spread or variability of data from the mean. Variance is the squared average distance from the mean, while standard deviation is the average distance. Smaller values indicate data points are closer to the mean. These measures are descriptive and do not imply any conclusions 05 AVERAGES MEAN, MEDIAN, AND MODE An average represents the central tendency of a set of values. The mean is calculated by adding all values and dividing by the number of values. The median is the middle value when the values are ordered, or the average of the two middle values if there is an even number of values. The mode is the value that appears most frequently in the dataset. 06 PARAMETER, STATISTICS, AND POPULATION Parameters are descriptive measures from a population, while statistics are from a sample. A finite population has a fixed number of elements, while an infinite population hypothetically has an endless number of elements. 07 SAMPLING DISTRIBUTION OF THE SAMPLE MEANS The sampling distribution of the sample means is a frequency distribution of means from all possible random samples of a specific size. It's crucial in statistical inference, enabling statisticians to infer population parameters from sample data and form the basis for confidence intervals and hypothesis tests. FORMULAS Discrete Probability Distribution 01 MEAN 02 VARIANCE (to get standard deviation, square root your variance) Sampling Distribution 01 MEAN (Sample mean = Population mean) 02 POPULATION VARIANCE (to get standard deviation, square root your variance) FINITE INFINITE 03 SAMPLE 04 SAMPLE (to get standard deviation, square root your variance) VARIANCE VARIANCE 05 COMBINATION Original code is nCr in calculator. Normal Distribution INDIVIDUAL SAMPLE 01 Z-SCORE 02 Z-SCORE Hello Learners. remember that this is made to simplify stuff and help you study for the exam. do not rely on this alone. open niyo parin courseware niyo ha -sir yow