Normal Distributions and Standard (z) Scores

Normal Distributions and Standard (z) Scores

•  Many real‒world phenomena like heights, weights, exam marks or daily temperatures tend to cluster around an average value. This pattern is called as normal distribution.

•  Definition: A normal distribution is a symmetrical, bell‒shaped curve that shows how data values are distributed around the mean(average).

•  The normal distribution is a continuous probability distribution. It is symmetric and bell shaped. It is also called as Gaussian distribution.

•  It describes how data is distributed around a center value or a mean value.

Characteristics

1) Symmetry: The distribution is perfectly symmetrical around the mean (average). The left and right sides of the curve are mirror images.

2) Mean, median and mode: In this distribution method, the Mean = Median = Mode.

3) Bell shaped curve: The graph of a normal distribution is a smooth, symmetric bell‒ shaped curve. The highest point of the curve is at the mean (μ). The total area under the Gaussian distribution curve equals 1. The normal distributions curve is unimodal (has one peak). The curve approaches the x‒axis but does not touch it.

4) Standard deviation :

• The spread or width of the distribution is determined by the standard deviation.

• About 68 % of the data falls within one standard deviation of the mean.

• About 95% of the data falls within two standard deviations of the mean.

• About 99.7% of the data falls within three standard deviations of the mean.

• This is called Empirical rule.

Normal distribution formula

Mathematically the Probability Density Function (PDF) of normal distribution is

where:

x = Value in the dataset

μ = Mean (average)

σ = Standard deviation (Spread of data)

e = Euler's number (≈ 2.718)

Example:1

A random variable X follows a normal distribution with mean of 10 and a standard deviation of 2. Find the value of the Probability Density Function (PDF) at x = 12.

Solution: The formula for finding PDF is

Given that:

(μ = 10, σ = 2, x = 12):

Simplify the exponent:

[(12‒10)² ] / 2.2²    = 4/8  =  0.5

The formula now becomes

f(12) =  [ 1 / 2√2π ] e ^‒0.5

√2π   ≈  2.5066,

e^‒0.5   ≈  0.6065.

f(12) = [1 / 2×2.5066 ] . 0.6065

          =  0.6065 / 5.0132   ≈ 0.1210

Hence, the value of the PDF at x = 12 is approximately 0.1210

Applications of normal distribution

1. Normal distribution is used in monitoring product dimensions or performance to ensure they meet specifications.

2.It is used for modeling stock returns, asset prices and risk analysis.

3.Normal distribution is used in analyzing heights, weights and other biological measurements.

4. Normal distribution is used for studying test scores, IQ and survey data.

Standard z‒score

• Sometimes we want to know how far a particular value is from the mean in terms of standard deviation.

• Definition of z‒score: The z‒score (or standard score) tells how many standard deviations a data point is above or below the mean.

Formula:

z =  (X ‒μ) / σ

where

X = Individual value

μ = Mean

σ = Standard deviation

Example:2

Consider Mean = 70, Standard deviation = 10, and student's score is 85. Calculate Z score and interpret your answer.

Solution :

The z score can be calculated as

z =  (X ‒μ) / σ = (85‒70) / 10

z = 1.5

Interpretation: The student scored is standard deviation above mean. That means this student has scored better than most of the students.

Example:3

Consider mean height = 165 cm, σ = 5 cm, X = 155 cm. Calculate z score and interpreter your result.

Solution :

z  = (155‒165) / 5  = ‒2

Interpretation: The height is 2 standard deviations below mean. So the height of student is short as compared to average.

Review Question

1. Why is the normal distribution called "bell‒shaped" ?