Data Handling and Probability

For FET Mathematics Teachers

Using the SHARP EL535HT scientific calculator.

What are we covering today?

• Data Handling or Statistics

– Central Tendency

– Variance and standard deviation

– Regression and correlation

– Interpolation and extrapolation

• Probability

– Relative frequency vs theoretical probability

– Fundamental Counting principle

But first a warm up

• Press

• Then

• You should see 4 modes

• Choose 2 for Drill

<MODE>0: NORMAL 1: STAT2: DRILL 3: TABLE

How Drill mode works

• 2 options:

– 0: MATH

• Math tests basic arithmetic skills (+, -, x, ÷, or + - x ÷ ). To select use and keys

• 25, 50 or 100 questions. To select use or

– 1: TABLE

• Tests a particular times-table (1 - 12). To select use or keys

• Serial or random. To select use or

Competition time!

• Heads up!

– Type in the answer and then press enter for the calculator to mark your answer

– If you get it right you will get a √ next to the question.

– If you get it wrong you will get a X and the same question will appear again

– If you accidently type in something wrong, press

to delete what you typed in.

First teacher to 100% wins the prize ☺

Data Handling – Central Tendency

• Mean, median and mode.

• Median – middle of the data

• Mode – happens the most

• Mean – is nasty to work out ☺

Example:

• E.g. 32 68 3 93 43 43 65

– Rearrange the data in ascending order:

• 3 32 43 43 65 68 93

– Median is in the middle so 43

– Mode – happens the most so also 43.

– Mean – the long way: ҧ𝑥 =σ 𝑥

𝑛

=347

7

= 49.57…

On the SHARP EL535HT Scientific Calculator

• Press 1 for STAT.

• Should see a menu like the one on the right.

• Choose SD for single data by pressing 1.

<< STAT-1 >>0: SD 1: LINE2: QUAD 3: E_EXP4:LOG 5: POWER

Now input the data:

• 3, 32, 43, 43, 65, 68, 93

• So press 3

• 32

• 43

• 65

• 68

• 93

93DATA

DATA SET = 7.

To find the mean:

• Press

• To show all working out steps:– Find the sum of x by

pressing:

– Find the number of observations by pressing:

ഥ𝒙 =

49.57142857

Variance and Standard Deviation

• Where it comes from:

– A deviation is the average distance of an observation from the mean.

• Because all the deviations add up to zero, we square the distance before adding them all up.

– The variance is the average of the squared distance from the mean.

– The standard deviation is thus the square-root of the variance.

The long version

𝒙 𝒙 − ഥ𝒙 𝒙 − ഥ𝒙 𝟐

3 -46,57 2 168.7649

32 -17,57 308.7049

43 -6,57 43.1649

43 -6,57 43.1649

65 15.43 238.0849

68 18.43 339.6649

93 43.43 1 886.1649

Total: 0.01 5 027.7143

Now find the average:

• So : Variance = 5 027.7143

7

= 718.2449

• Square-root the above value to find the standard deviation

– Standard deviation = 718.2449

= 26.80…

On the SHARP EL535HT Scientific Calculator

• To find the standard deviation:

– Press

• And that’s it ☺

• To find the variance

– press

𝜎𝑥2 =

718.244898

Regression and Correlation

• What is it?

– The regression line gives the best fit of a straight line for the given data

• This means that we need an x and a corresponding y

– A correlation shows how much one set of values matches its corresponding set of values

• Does the one have a direct and strong effect on the its matching value.

An example:

X Y

5 42

4 46

2 6

5 77

3 22

2 2

1 1

6 85

5 84

Draw a scatterplot of the data

Now input the data:

• So press 5 42 • 4 46• 2 6• 5 77• 3 22• 2 2• 1 1• 6 85• 5 84

5,84DATA

DATA SET = 9.

On the SHARP EL535HT Calculator

• To find the gradient:

– Press

• To find the y-intercept

– Press

• To find the correlation coefficent

– Press

r=

0.934862118

Interpolation and Extrapolation

• What is it?

– Interpolation is estimating a value for a given x or y within the given data set

– Extrapolation is estimating a value for a given x or y outside of the maximum and minimum values for x and y.

• Generally substitute the x or y value into the regression equation but….

We can also find it on the Sharp El535HT

• If we are given that x = 4

– Press 4

• If we are given that y = 46.85

– Press 46.85

• To clear all the data stored press

4y’

46.85648148

Probability

Relative Frequency Vs Theoretical Probability

• What is relative frequency– How often one thing happens (e.g. rolling a 1 on a

die) in comparison with how many times the general thing happens (e.g. rolling the die).

• What is theoretical probability – How many times that one thing should happen

out of the general thing according to theory (usually given as a fraction or a ratio).

– E.g. for every 6 rolls of the die, 1 should technically or theoretically show up once.

Random Mode

• On your calculator press:

– 0

• There is a random mode on your calculator. To get to it press

NORMAL MODE

0.

You should see the following options:

• 0: RAND → random decimals (3 decimal places)

• 1: R-DICE → numbers 1 to 6 like a die

• 2: R-COIN → numbers 0 and 1 – choose which is heads and tails

• 3: R-INT → random numbers between 0 and 99.

• Press 1 and to roll the die

<RANDOM>0: RAND 1: R-DICE2: R-COIN 3: R-INT

Class Exercise

• relative frequency

• And cumulative frequency

More Examples of Cumulative Frequency

• The chance of picking the right lotto numbers:– Can choose from 49 possible numbers:

– So the chance of getting the first number is 1

49

– The chance of getting the second number is 1

48

– And so on…

– So if we can choose 5 numbers we have:

•1

49×

1

48×

1

47×

1

46×

1

45= 0. 000000004 chance of getting

5 numbers correct.

• Or a 1 in 250 000 000 chance of getting 5 numbers correct.

What’s the chance of not winning?

• Well if we pick 5 numbers– The chance of not picking the first number correctly is

48

49

– The chance of not picking the second number

correctly is 47

48and so on

• So now we have 48

49×

47

48×

46

47×

45

46×

44

45=

44

49

• Or 0.897959183 or approximately 90%

• Note: the other 10% is the other possible combinations of number choices.

Fundamental Counting Principle

• What is it?

– It gives you a way to work out how different things can be arranged and in how many different ways they can be arranged.

– It includes permutations and combinations

Using students in class as an example

• Ask for 5 volunteers and have them stand in front of the class.

• Ask them in how many possible ways you can arrange the 5 volunteers

• Have the volunteers arrange themselves in all the different combinations they can think of.

• Wait for them to figure out its going to take a long time to find out all the different combinations.

• Start by having the first volunteer stand in the first space in the line.

• And the second volunteer stand next to them.

• Now there are only 3 students to arrange in different spaces – they should quickly realisethere are only 6 ways to now organisethemselves.

• Ask them why.

• This is because there are 3 options for the first space, 2 options for the second space and 1 for the last space. Or 3 x 2 x 1 = 6

• Now think about the 4 volunteers – there are 4 options for the first position, 3 for the second and so on. Or 4 x 3 x 2 x 1 = 24

• And then for 5 volunteers – 5 x 4 x 3 x 2 x 1 = 120

• Which leads into factorials.

Factorials

• For the previous example we would like 5!

– So press 5

5!=

120.

Permutations

• Permutations are a set of items that are arranged in a specific order, specifically where there are more items than spaces available.

• For example – in a class of 20 students the teacher gives a prize to the top 3 students. How many different ways can the prizes be won?

On the SHARP EL5355HT calculator

• n =20 and r = 3

• So press: 20

3

20P3=

6’840.

Permutations Summary

• Order matters!

• No repeated options use

– Formula: 𝑛!

𝑛−𝑟 !

– Or on the calculator n r

• Repeated options:– All values can be repeated: use 𝑛𝑟

– Only certain values are repeated: use permutation formula and then divide the answer by each repeat that has been factorialised.

Combinations

• In combinations – order doesn’t matter.

• There are two types:

– 1: Where all items have a place

– 2: Where not all the items have a place

On the SHARP EL535HT Calculator

• Example: In a class of 20 students 3 students are picked randomly from the class.

• So type: 20

3

20C3=

1’140.

Summary for Combinations

• In combinations – order doesn’t matter!

• That means you have less combinations.

• No repetition

– Use the formula: 𝑛!

𝑟! × 𝑛 −𝑟 !

– Or use the calculator: n r

• With repetion:

– Can only use the formula: 𝑛+𝑟 −1 !

𝑛−1 !𝑟!

Thank you!

For free worksheets with memos, and past exams please visit

www.mathsatsharp.co.za