Data Science Normal Distribution Z-Score
- 1. Data Science - Normal Distribution Dr.M.Pyingkodi Associate Professor Dept. of MCA Kongu Engineering College Erode, Tamil Nadu,India
- 2. • A Normal Distribution • A Normal Distribution is a bell-shaped curve that is symmetric about the mean, where most of the data points are clustered around the center and fewer are found as you move away from the mean. • Characteristics • 1. Symmetry • The distribution is symmetric about the mean. • This means that the left side of the distribution is a mirror image of the right side. • 2. Bell-Shaped Curve • The graph of a normal distribution is shaped like a bell, with a single peak at the mean. • It rises gradually on both sides and tails off symmetrically. • 3. Mean, Median, and Mode Equality • In a normal distribution, the mean, median, and mode are all equal and located at the center of the distribution. • This central point is also known as the peak of the curve. Dr.M.Pyingkodi, Associate Professor, Dept of MCA , Kongu Engineering College, Tamil Nadu, India 2
- 3. • A Normal Distribution • 4. 68-95-99.7 Rule • Approximately 68% of the data falls within one standard deviation of the mean, about 95% falls within two standard deviations, and about 99.7% falls within three standard deviations. • This is often referred to as the empirical rule. • 5. Asymptotic Nature • The tails of the normal distribution approach, but never actually touch, the horizontal axis. • This means that there is a theoretical possibility of observing values far from the mean, though these occurrences become increasingly rare. Dr.M.Pyingkodi, Associate Professor, Dept of MCA , Kongu Engineering College, Tamil Nadu, India 3
- 4. z-score • A z-score is a statistical measure that indicates how many standard deviations a data point is from the mean of a dataset. • It is a way to standardize scores from different distributions, allowing for comparison. What a Z-Score Indicates ? • 1.Position Relative to the Mean • A z-score tells you how far away a specific value (data point) is from the mean. A positive z-score indicates that the data point is above the mean. A negative z-score indicates that the data point is below the mean. A z-score of 0 means the data point is exactly at the mean. Dr.M.Pyingkodi, Associate Professor, Dept of MCA , Kongu Engineering College, Tamil Nadu, India 4
- 5. • 2. Standard Deviations • The value of the z-score represents the number of standard deviations the data point is from the mean. • For example, a z-score of 2 means the data point is two standard deviations above the mean, while a z-score of -1.5 means • it is one and a half standard deviations below the mean. • Example • Consider a dataset of test scores for a class, with the following characteristics: • • Mean (μmuμ) - 70 • Standard Deviation (σsigmaσ) -10 • Suppose a student scores 85 on the test. • To calculate the z-score for this student's score, we use the formula • Where • X = the data point (student's score) • m= the mean • σ = the standard deviation Dr.M.Pyingkodi, Associate Professor, Dept of MCA , Kongu Engineering College, Tamil Nadu, India 5
- 6. • Substituting in the values • In this example, the student's z-score is 1.5. • This means that the student scored 1.5 standard deviations above the mean score of the class. • This indicates a relatively high performance compared to the average student in the class. • Using z-scores allows us to understand the position of a data point within the context of the overall dataset, making it easier to compare scores across different distributions. • Why Z-Score ? • Comparison Across Different Datasets • Z-scores allow us to compare values from different datasets, even if they have different units or scales. • Dr.M.Pyingkodi, Associate Professor, Dept of MCA , Kongu Engineering College, Tamil Nadu, India 6
- 7. • For instance, if you're comparing the performance of two students on different tests, the z-scores normalize their scores, making it easier to compare their relative performance. • Probability and Percentiles • In a normal distribution, z-scores are tied to probabilities. • For example, a z-score of +1 corresponds to about the 84th percentile of the data, meaning that 84% of the data points are below this score. • Identifying Outliers • Z-scores can highlight extreme values. If a z-score is very high or low (e.g., beyond ±3), the value may be considered an outlier in the dataset. • Imagine the average height of adult men in a population is 175 cm with a standard deviation of 10 cm. • This means he is 1.5 standard deviations taller than the average • Z-scores make it easier to understand how unusual or typical a value is relative to the overall data distribution. Dr.M.Pyingkodi, Associate Professor, Dept of MCA , Kongu Engineering College, Tamil Nadu, India 7