Finding Sigma and Mu

How do I find the mean (μ) or the standard deviation (σ) if one of them is unknown?

If the mean or standard deviation of the $X ~ N (μ, σ^{2})$ is unknown then you will need to use the standard normal distribution
You will need to use the formula
- $z = \frac{x - μ}{σ}$ or its rearranged form $x = μ + σ z$
You will be given a probability for a specific value of $x (P (X < x) = p or P (X > x) = p)$
To find the unknown parameter:
STEP 1: Sketch the normal curve
- Label the known value and the mean
STEP 2: Find the z-value for the given value of $x$
- Use the table of the Normal Distribution to find the value of z such that $P (Z < z) = p$ or $P (Z > z) = p$
- Make sure the direction of the inequality for $Z$ is consistent with $X$
- The table gives the $z$ -value to three decimal places to avoid rounding errors
  - Use the sketch to help you decide whether your z value is positive or negative
  - You should use the 3 decimal places throughout your calculations so that your final answer can be rounded to 3 significant figures
STEP 3: Substitute the known values into $z = \frac{x - μ}{σ}$ or $x = μ + σ z$
- You will be given x and one of the parameters (μ or σ) in the question
- You will have calculated z in STEP 2
STEP 4: Solve the equation

How do I find the mean (μ) and the standard deviation (σ) if both of them are unknown?

If both of them are unknown then you will be given two probabilities for two specific values of $x$
The process is the same as above
- You will now be able to calculate two z-values
- You can form two equations (rearranging to the form $x = μ + σ z$ is helpful)
- You now have to solve the two equations simultaneously (you can use your calculator to do this)
- Be careful not to mix up which z-value goes with which value of $x$

Worked example

It is known that the times, in minutes, taken by students at a school to eat their lunch can be modelled using a normal distribution with mean μ minutes and standard deviation σ minutes.

Given that 10% of students at the school take less than 12 minutes to eat their lunch and 5% of the students take more than 40 minutes to eat their lunch, find the mean and standard deviation of the time taken by the students at the school.

3-3-4-finding-mu-and-sigma-we-solution

Examiner Tip

These questions are normally given in context so make sure you identify the key words in the question. Check whether your z-values are positive or negative and be careful with signs when rearranging.

Finding Sigma and Mu (CIE AS Maths: Probability & Statistics 1)

Revision Note

How do I find the mean (μ) or the standard deviation (σ) if one of them is unknown?

How do I find the mean (μ) and the standard deviation (σ) if both of them are unknown?

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1. Data Presentation & Interpretation

1.1 Statistical Measures

1.1.1 Basic Statistical Measures

1.1.2 Frequency Tables

1.1.3 Standard Deviation & Variance

1.1.4 Coding

1.2 Representation of Data

1.2.1 Data Presentation

1.2.2 Stem and Leaf Diagrams

1.2.3 Box Plots & Cumulative Frequency

1.2.4 Histograms

1.3 Working with Data

1.3.1 Interpreting Data

1.3.2 Skewness

2. Probability

2.1 Basic Probability

2.1.1 Calculating Probabilities & Events

2.1.2 Venn Diagrams

2.1.3 Tree Diagrams

2.2 Permutations & Combinations

2.2.1 Arrangements & Factorials

2.2.2 Permutations

2.2.3 Combinations

2.3 Further Probability

2.3.1 Set Notation & Conditional Probability

2.3.2 Further Tree Diagrams

2.3.3 Further Venn Diagrams

2.3.4 Probability Formulae

3. Statistical Distributions

3.1 Probability Distributions

3.1.1 Discrete Probability Distributions

3.1.2 E(X) & Var(X) (Discrete)

3.2 Binomial & Geometric Distribution

3.2.1 The Binomial Distribution

3.2.2 Calculating Binomial Probabilities

3.2.3 The Geometric Distribution

3.3 Normal Distribution

3.3.1 The Normal Distribution

3.3.2 Standard Normal Distribution

3.3.3 Normal Distribution - Calculations

3.3.4 Finding Sigma and Mu

3.4 Working with Distributions

3.4.1 Modelling with Distributions

3.4.2 Normal Approximation of Binomial

Author: Amber