Properties of the Normal Distribution

The normal distribution, also known as the Gaussian distribution, is a cornerstone of statistics and probability. Its widespread use stems from its distinctive mathematical properties and its remarkable tendency to appear in real-world data.

Here are the essential properties of a normal distribution:

1. Equality of Mean, Median, and Mode

In a perfectly normal distribution, the three measures of central tendency coincide:

$$ \text{Mean} = \text{Median} = \text{Mode} $$

This equality signifies perfect symmetry. The single peak of the distribution represents the most frequent value (mode), the middle value when data is ordered (median), and the average of all values (mean).

2. Always Positive Probability Density

The probability density function (PDF) of a normal distribution, denoted as $f(x)$, is always greater than zero for any real number $x$:

$$ f(x) > 0 \quad \text{for all } x \in (-\infty, \infty) $$

This means that every possible value on the number line, no matter how extreme or far from the mean, has a non-zero (though potentially very small) probability of occurring.

3. Defined by Two Parameters: Mean ($\mu$) and Standard Deviation ($\sigma$)

A normal distribution is uniquely characterized by two parameters:

• Mean ($\mu$): This parameter determines the center or location of the distribution's peak on the horizontal axis.
• Standard Deviation ($\sigma$): This parameter dictates the spread or width of the distribution. A smaller $\sigma$ results in a narrower, more peaked curve, while a larger $\sigma$ leads to a wider, flatter curve.

These two parameters, $\mu$ and $\sigma$, completely define the shape and position of the normal distribution curve, resulting in a uni-modal, bell-shaped, and symmetrical distribution.

4. Symmetrical About the Mean

The normal distribution curve is perfectly symmetrical around its mean ($\mu$). This symmetry can be expressed mathematically as:

$$ f(\mu - x) = f(\mu + x) $$

This implies that the left side of the curve is a mirror image of the right side. Consequently, data values that are equidistant from the mean have the same probability density.

5. Total Area Under the Curve is Equal to 1

The total area enclosed by the normal distribution curve and the horizontal axis represents the sum of all possible probabilities, which must equal 1 (or 100%):

$$ \int_{-\infty}^{\infty} f(x) , dx = 1 $$

This property is fundamental, as it confirms that the distribution accounts for all possible outcomes.

6. Equal Distribution on Both Sides of the Mean

Due to its perfect symmetry, exactly 50% of the data values in a normal distribution lie to the left of the mean, and 50% lie to the right:

$$ P(X < \mu) = P(X > \mu) = 0.5 $$

This means that the mean also serves as the median, splitting the probability mass equally.

7. Uni-modal Curve

A normal distribution possesses a single peak, which occurs at the mean ($\mu$). This characteristic defines it as a uni-modal distribution. There are no secondary peaks or modes in a standard normal distribution.

Summary of Properties

---

SEO Keywords

Normal distribution properties, Mean median mode equality, Positive probability density normal curve, Normal distribution parameters $\mu$ and $\sigma$, Symmetry of normal distribution, Total area under normal curve, Probability distribution equal halves, Uni-modal normal distribution, Bell-shaped curve characteristics, Statistical properties of Gaussian distribution.