Normal Distribution EXPLAINED with Examples

The normal distribution is a symmetric, bell-shaped curve, centered around the mean. Probability decreases as you move away from the mean, with significant values typically within three standard deviations. The 68-95-99.7 rule states that 68% of data falls within one standard deviation, 95% within two, and 99.7% within three standard deviations of the mean.

The normal distribution is highly applicable to real-world phenomena like heights, weights, and IQ scores. To compare different distributions with varied units and scales, data is standardized using a z-score. Standardization transforms a distribution to have a mean of 0 and a standard deviation of 1, allowing for consistent analysis.

A z-score measures how many standard deviations a data point is from the mean, calculated by (data point - mean) / standard deviation. An example with people's heights demonstrates how an unstandardized distribution can be converted to a standardized one using z-scores, allowing for a universal comparison.

Unlike other distributions, normal distribution probabilities are typically found using z-charts. These charts map z-scores to their associated probability values, specifically representing the area or probability to the left of a given z-score. It's crucial to use the correct chart based on whether the z-score is positive or negative.

Using an example of pizza sizes with a mean of 16.3 inches and a standard deviation of 0.2 inches, the video demonstrates how to calculate the probability of getting a free pizza (less than 16 inches). This involves calculating the z-score for 16 inches, which is -1.5, and then using the z-chart to find the corresponding probability of 0.0668, or a 6.68% chance.

To find the probability of a pizza being over 16.5 inches, first calculate the z-score, which is 1.0. The z-chart provides the probability to the left (0.8413). To find the probability to the right (over 16.5 inches), subtract this value from 1, resulting in 0.1587, or a 15.87% chance.

Finally, the video covers finding the probability of a pizza being between 15.95 and 16.63 inches. This involves calculating z-scores for both values (-1.75 and 1.65, respectively). The probability between these two points is found by subtracting the probability to the left of the smaller z-score from the probability to the left of the larger z-score. This yields 0.9505 - 0.0401 = 0.9104, or a 91.04% chance.