The standard deviation of the normal distribution is a measure of how spread out the distribution is from its mean. It is the square root of the variance of the distribution. Here are some key points about standard deviation:
- The standard deviation can be represented with the Greek letter sigma (σ).
- A normal distribution with a small standard deviation will have most of its data points close to the mean, while a distribution with a large standard deviation will have data points that are more spread out.
- About 68% of the data in a normal distribution falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.
- The standard deviation can be used to calculate confidence intervals, or ranges of values that are likely to contain the population parameter being estimated.
- The standard deviation is influenced by outliers in the data, so it may not be the most appropriate measure of dispersion in skewed or non-normal distributions.
Overall, the standard deviation is a useful measure of how tightly or loosely clustered data is around the mean in a normal distribution, and can be used to make inferences about the population based on sampled data.
Example problem for standard deviation of normal distribution
What is the formula for standard deviation of normal distribution?
- The formula for standard deviation of normal distribution is:
- σ = √( Σ(xi – μ)^2 / N )
What does standard deviation of normal distribution represent?
- Standard deviation of normal distribution represents the amount of variation or dispersion of a set of data from the mean.
How does the standard deviation affect the normal distribution curve?
- A smaller standard deviation will result in a narrower and taller normal distribution curve, while a larger standard deviation will result in a flatter and wider normal distribution curve.
What is the relationship between standard deviation and the empirical rule?
- The empirical rule states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% of the data falls within two standard deviations of the mean, and 99.7% of the data falls within three standard deviations of the mean.