Normal Distribution Concepts and Properties (2024)

To find the areas of the regions under the normal curve that correspond to the ______ value, simply find the area of the given z-value using the z-Table.

The area that corresponds to z = 0.60 is found at the intersection of ______ and .00 in the z-Table.

The z-Table is also known as the Table of ______ under the Normal Curve.

The area that corresponds to z = -1.47 is ______.

The z-score is also known as the ______ score.

To find the area that corresponds to a given z-score, one needs to express the z-score into ______ decimal form.

The area under the standard normal curve can be found between ______ and ______ values of z-score.

The z-Table is used to find the area that corresponds to a given ______ value.

In 1733, Abraham de Moivre first discovered the ______ distribution.

The normal distribution refers to a ______ probability distribution.

The tails of the curve flatten out indefinitely along the ______ axis.

The total area under the normal curve is equal to ______ or 100%.

The normal curve is often called the ______ Distribution.

A random variable X whose distribution has the shape of a ______ curve is called a normal random variable.

A change in the value of the ______ shifts the graph of the normal curve to the right or left.

The normal distribution has a ______ peak.

A large standard deviation means that the distribution is rather ______ out, with some chance of observing values at some distance from the mean.

The curve is ______ about its center.

The standard normal curve is a normal probability distribution that has a ______ µ=0 and a standard deviation σ=1 unit.

The mean, median and mode coincide at the ______.

About ______ of the area under the curve falls within 1 standard deviation from the mean.

The Empirical Rule is also called as the ______ Rule.

The width of the curve is determined by the ______ deviation of the distribution.

The standard normal curve is a normal probability distribution that has a mean µ=0 and a standard deviation σ=______ unit.

The probability notation P ( z < a ) reads as the probability of z being ______ than a.

The probability notation P ( z > a ) reads as the probability of z being ______ than a.

The probability notation P ( a < z < b ) reads as z is ______ a and b.

In Example 1, the probability of the area below z = 0.50 is ______.

In Example 2, the probability of the area that is at least z = -2 is ______.

In Example 3, the area of interest is between z = -1.5 and z = ______.

To find the probability of the area below z = 0.50, the z-Table is consulted to find the area that corresponds to z = ______.

A ______ score is a measure of relative standing that tells how many standard deviations either above or below the mean a particular value is.

In Example 2, the probability notation P (z > -2.00) is equal to 1 - P(z ______ -2).

The scores represent the distances from the center measured in ______ units.

The areas under the normal curve are given in terms of ______ values or scores.

The raw score X is above the mean if ______ is positive and it is below the mean when ______ is negative.

X = the given measurement of a normal random _______,

µ represents the population ______.

σ represents the population ______ standard deviation.

The z-score is also known as the STANDARD ______.

Introduction to Normal Distribution

• The normal distribution was first discovered by Abraham de Moivre in 1733.
• It is also known as the Gaussian distribution or bell-shaped curve.
• The normal distribution plays a crucial role in inferential statistics.

Definition of Normal Distribution

• A random variable X is said to be normally distributed with mean µ and standard deviation σ.
• The normal distribution refers to a continuous probability distribution described by the normal equation.

Properties of Normal Distribution

• The distribution curve is bell-shaped and has a single peak, making it unimodal.
• The curve is symmetrical about its center.
• The mean, median, and mode coincide at the center.
• The width of the curve is determined by the standard deviation of the distribution.
• The tails of the curve flatten out indefinitely along the horizontal axis, but never touch it.
• The total area under the normal curve is equal to 1 or 100%.

Factors Affecting the Graph of Normal Distribution

• Mean determines the location of the center of the bell-shaped curve.
• Standard deviation determines the shape of the graph, particularly the height and width of the curve.

Standard Normal Curve

• A standard normal curve is a normal probability distribution with a mean µ=0 and a standard deviation σ=1 unit.
• Standardizing the normal curve makes it easier to work with and allows for the transformation of observations of any normal random variable X to a new set of observations of another normal random variable Z with mean 0 and standard deviation 1.

Empirical Rule (68-95-99.7 Rule)

• About 68.26% of the area under the curve falls within 1 standard deviation from the mean.
• About 95.44% of the area under the curve falls within 2 standard deviations from the mean.
• About 99.74% of the area under the curve falls within 3 standard deviations from the mean.

Areas under the Normal Curve

• To find the areas of the regions under the normal curve, use the z-Table (Table of Areas under the Normal Curve).
• Examples of finding the areas that correspond to given z-score values are provided.

Standard Score (Z-Score)

• A z-score is a measure of relative standing that tells how many standard deviations above or below the mean a particular value is.
• The z-score represents the distances from the center measured in standard deviation units.
• Importance of z-score: raw scores may be composed of large values, but these large values cannot be accommodated at the baseline of the normal curve, so they need to be transformed into scores for convenience without sacrificing meanings associated with the raw scores.

Applications of Normal Distribution

• Probability notations under the normal curve: P(z < a) less than z, P(z > a) greater than z, and P(a < z < b) z is between a and b.
• Examples of finding the probability of the area below z = 0.50, the area that is at least z = -2, and the area between z = -1.5 and z = 2 are provided.