Normal Distribution and Sampling Distribution of Means

Probability Density Function

The probability density function (PDF) is a fundamental concept in statistics used to compute probabilities for continuous random variables. It possesses two essential properties:

• Non-negativity: The graph of the PDF must lie on or above the horizontal axis.

• Total Area: The area under the graph must equal 1, representing the total probability.

Example: The normal distribution is a common example of a probability density function.

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its bell-shaped and symmetric curve. It is also known as the Gaussian Distribution, named after Carl Gauss, and is widely used to model natural phenomena.

• Notation: Normal distributions are denoted as N(μ, σ) or N(μ, σ2), where μ is the mean and σ is the standard deviation.

• Range: The variable x ranges from −∞ to +∞.

• Parameters: The distribution is fully described by its mean (μ) and standard deviation (σ).

Example: N(20, 3) or N(20, 32) represents a normal distribution with mean 20 and standard deviation 3.

Features of the Normal Curve

The normal curve exhibits several important features:

• Bell-shaped: The curve is highest at the mean and symmetric about a vertical line through μ.

• Central tendency: The mean, median, and mode are approximately equal (μ ≅ median ≅ mode).

• Asymptotic: The curve approaches but never touches the horizontal axis.

• Spread: As σ increases, the curve spreads out; as σ decreases, it becomes more peaked around μ.

Example: Curves with the same mean but different standard deviations will have different spreads.

Normal Distribution Function

The mathematical formula for the normal distribution is:

• Where μ is the population mean and σ is the population standard deviation.

Example: A curve with μ = 2 and σ = 0.25 is more peaked than one with μ = 3 and σ = 0.5.

Normal Probability and Area Under the Curve

The area under the normal curve within a given interval represents the probability that a measurement will fall within that interval. The total area under the curve is 1, and the data is equally distributed on each side of the mean:

• 50% of the data lies to the left of μ

• 50% of the data lies to the right of μ

Identifying Normal Data

There are several methods to determine if a dataset follows a normal distribution:

• Histogram: A normal distribution's histogram should be roughly bell-shaped.

• Outliers: A normal distribution should have no more than one outlier.

• Quantile Plot (QQ Plot): If the data points lie close to a straight line, the data is approximately normal.

Example: The histogram below shows a bell-shaped distribution, indicating normality.

Empirical Rule (68-95-99.7 Rule)

The empirical rule describes the proportion of data within certain standard deviations of the mean in a normal distribution:

• About 68% of the area is within one standard deviation (μ ± σ)

• About 95% of the area is within two standard deviations (μ ± 2σ)

• About 99.7% of the area is within three standard deviations (μ ± 3σ)

This rule is useful for quickly estimating probabilities and identifying outliers in normally distributed data.