Start studying Search for a subject Log in Join now
A Level Maths: Probability & Statistics 1 CIE Revision Notes 3. Statistical Distributions 3.3 Normal Distribution 3
The Normal Distribution (CIE A Level Try premium for free
Maths: Probability & Statistics 1) Reach your learning goals faster with
unlimited access to Revision Notes
Revision Note Start your 7-day free trial
Download PDF Test yourself
1. Data Presentation & Interpret…
Author Expertise
Dan Maths
2. Probability
3. Statistical Distributions
3.1 Probability Distributions
3.2 Binomial & Geometric …
3.3 Normal Distribution
05:09
3.3.1 The Normal Distribution
3.3.2 Standard Normal Distribution
Did this video help you? Yes No
3.3.3 Normal Distribution -
Calculations
3.3.4 Finding Sigma and Mu
Unlock unlimited access to 150k+
examiner-written resources 3.4 Working with Distributi…
Start your 7-day free trial
Properties of Normal Distribution
The binomial distribution is an example of a discrete probability distribution. The normal
distribution is an example of a continuous probability distribution.
What is a continuous random variable?
A continuous random variable (often abbreviated to CRV) is a random variable that
can take any value within a range of infinite values
Continuous random variables usually measure something
For example, height, weight, time, etc
What is a continuous probability distribution?
A continuous probability distribution is a probability distribution in which the random
variable X is continuous
The probability of X being a particular value is always zero
P(X = k) = 0 for any value k
Instead we define the probability densityDownload x) foron
function f(notes 3.3.1 Thevalue
a specific Normal Distribution
We talk about the probability of X being within a certain range
A continuous probability distribution can be represented by a continuous graph (the
values for X along the horizontal axis and probability density on the vertical axis)
The area under the graph between the points x = a and x = b is equal to P(a ≤ X ≤ b)
The total area under the graph equals 1
As P(X = k) = 0 for any value k , it does not matter if we use strict or weak inequalities
P(X ≤ k) = P(X < k) for any value k
What is a normal distribution?
A normal distribution is a continuous probability distribution
The continuous random variable can follow a normal distribution if:
The distribution is symmetrical
The distribution is bell-shaped
If X follows a normal distribution then it is denoted X ∼ N (μ , σ 2)
μ is the mean
σ2 is the variance
σ is the standard deviation
If the mean changes then the graph is translated horizontally
If the variance changes then the graph is stretched horizontally
A small variance leads to a tall curve with a narrow centre
A large variance leads to a short curve with a wide centre
What are the important properties of a normal distribution?
The mean is μ
The variance is σ2
If you need the standard deviation remember to square root this
The normal distribution is symmetrical about x = μ
Mean = Median = Mode = μ
The normal distribution curve has two points of inflection
x = μ ± σ (one standard deviation away from the mean)
There are the results:
Approximately two-thirds (68%) of the data lies within one standard deviation
of the mean (μ ± σ)
Approximately 95% of the data lies within two standard deviations of the mean
(μ ± 2σ)
Nearly all of the data (99.7%) lies within three standard deviations of the mean
(μ ± 3σ)
For any value x a z-score (or z-value) can be calculated which measures how many
standard deviations x is away from the mean
x −μ
z =
σ
Only 2 pages have been converted.
Please go to https://docs.zone and Sign Up to convert all pages.
