1. M EAN & V ARIANCE OF B INOMIAL D ISTRIBUTIONS AS Maths with Liz Stats 1

2. AIMS By the end of the lesson, you should be able to… Find the mean or expected value of a binomial distribution. Find the variance and standard deviation of variance. P RIOR K NOWLEDGE Binomial distribution & notation involved Basic definitions of mean, variance, & standard deviation

3. B ASIC D EFINITIONS RECAP The mean measures the central tendency of the data. It can be found by adding up all of the data and dividing by the total number of data points there are. Notation: stands for the mean Greek letter mu (“mew”) that also stands for the mean the sum of all data divided by the number of data points in the set

4. B ASIC D EFINITIONS RECAP Variance works hand in hand with standard deviation to measure how close, on average, the data points are to the mean, and, in turn, tell you how reliable the mean actually is. It measures the spread of the data. It can be found by… subtracting each data point from the mean squaring it adding up each value you get from the above calculations and dividing this by the total number of data points in the set Notation: stands for the variance Greek letter sigma means sum or add up square of the difference of each data point and the mean divided by the number of data points in the set

5. B ASIC D EFINITIONS RECAP Standard deviation works hand in hand with variance to measure how far, on average, the data varies from the mean. If there is a low standard deviation, the data is considered to be consistent. It can be found by taking the square root of the variance. Notation: stands for the standard deviation the sum of the square of each data point the square of the mean divided by the number of data points in the set Another useful form (sometimes a shortcut):

6. M EAN OF B INOMIAL D ISTRIBUTIONS For a binomial distribution, the mean (also known as the “expected value” is given by “mean”: “expected value”: These are given in your formula booklet on page 11.

7. V ARIANCE OF B INOMIAL D ISTRIBUTIONS For a binomial distribution, the variance is given by This is also given in your formula booklet on page 11. S TANDARD D EVIATION OF B INOMIAL D ISTRIBUTIONS For a binomial distribution, the standard deviation is given by NOT GIVEN in formula booklet! Just remember, s.d. is always the square root of the variance.

8. E XAMPLE 1: Find the expected value, variance, and standard deviation for X given Mean:Variance:Standard Deviation

9. Y OU TRY ! Find the expected value, variance, and standard deviation for X given Mean:Variance:Standard Deviation

10. Y OU TRY ! Solution:

11. E XAMPLE 2

12. E X. 2 - S OLUTION ( PART A ) If Then Hint: You can always use your distribution table to work this out also!

13. E X. 2 – S OLUTION (PART B) If

14. E X. 2 – S OLUTION (PART C) Therefore, So, Keep in mind, this only represents 5 days (1 work week) and the probability is also in DAYS To calculate 52 weeks, we simply need to work out the expected value when This represents 52 work weeks, and the probability is in WEEKS He can only expect the his train to be on time all 5 days for approximately 12 weeks in the year (out of 52 total). CRAZY!

15. I NDEPENDENT S TUDY Stats Textbook: Pg. 84, Ex. 3E DUE FRIDAY, MARCH 20 TH O PTIONAL P RACTISE mymaths – expectation and variance MEI Worksheet “AQA Stats 1, Section 2: Using the Binomial Distribution” (on my moodle page under today’s date)