Harry Potter Guide to s 1

www.drfrostmaths.com

The Harry Potter Guide To S1 Last updated – 14th April 2015

Chapter 0 - Guide To Using Your Casio Calculator _

Practice being able to calculate these with your calculator - you should not

not rely on them to calculate your answer in the first place, but allows you to

check if you've got your answer correct. We'll use the following data sets:

Lengths of wands: 12.3cm, 15.7cm, 20.4cm, 21.3cm, 29.2cm

Weights of owls (kg) Frequency

Hours Revised Potions test score

Harry Tip #1: Be careful to note how many variables there are in the problem.

The first data set is obviously just one variable (time). The second table (where the data is grouped)

is still one variable, but has frequency information. On your calculator, press MODE then choose 2

for STAT, and finally 1-VAR for "1 variable". The get the frequency column, go to SETUP (SHIFT ->

MODE), press the down key to get to next page on the menu, choose 3 for STAT, then finally select 1

for 'FREQUENCY ON'. You may as well leave this mode on, since the frequency value defaults to 1

when you don't specify it (i.e. you just have one instance of each listed piece of data).

For the second data set, you must not choose the 'A + BX' mode and use your second Y column for

the frequencies, as you can't treat the frequency data as just a second variable.

For the last data set, since there's two variables, from SETUP -> STAT, choose 2 for 'A + BX' for linear

regression with two variables. This is the equation of the straight line, and this is because for the

purposes of S1, your variables are assumed to have a linear relationship (i.e. roughly follow a line of

best fit) when calculating both your Product Moment Correlation Coefficient (which measures how

well your data fits a straight line) and your regression line. The other modes, which you won't use at

A Level, allows your variables to have different relationships, e.g. for population growth (which

grows/falls exponentially), the 'A.B^X' mode would be more appropriate as is an

exponential function relating the two variables and .

Entering your data

Enter each value for your first variable, pressing = each time to get to the next row. If you have a

second variable or frequencies to enter, use the arrow keys to navigate back to the top right of your

table. Once done, press the AC key to 'bank' your table. It's now stored in memory.

Calculating a statistic

Press SHIFT -> 1 for 'STATISTIC'. Choose 'VAR' for calculate either the mean or variable of either of

your variables ( and where relevant ). This will then use this quantity in your current calculation,

which you can further manipulate if you like. Then press = to get the value.

Here's a summary of what you can calculate:

"In terms of wizarding

prowess, you're three

standard deviations above

the mean, Harry."

Set 1

Set 2

Set 3


VAR Standard deviation (don't use ), number of data values , and the mean .

SUM The sum of the values of your variable , or the sum of the squares , or the sum of the products . The second and third of these are obviously crucial when you're calculating , and , which your calculator is unable to

do directly.

Important note: Note that when you have a frequency row/column, is actually calculating , and is actually calculating because the values are being duplicated so each copy of the value is effectively being treated as a separate value of . We'll discuss this later.

REG Only available when you're in 2-variable mode. Allows you to calculate the y-intercept and gradient of your line of best fit, and in and the Product Moment Correlation Coefficient .

Now have a go at calculating the following for the 3 data sets above:

Data Set 1 The mean, variance and standard deviation of the length of wands.

Answers: , ,

Data Set 2 An estimate of the mean and standard deviation of the weight (recall that you need to use the midpoints of the class intervals).

Try calculating the standard deviation by typing

in the full formula for variance,

by

repeated use of the 'STATISTIC' button before pressing =. Verify that this gives you the same value as when you use your calculator to calculate directly.

,

Data Set 3 Try and find , , , and . Hence find , and using these values. Find

directly and the linear regression line.

, , , ,


Chapters 2/3 - Data: Location and Spread _

Hagrid Tip #1: When you have a discrete list of items, to find the median/quartiles, find

,

,

of the number of items , and then round up and use that numbered item. The one

exception is when you have a whole number after dividing, in which case use this item and

the one after.

Example: 2, 4, 6, 8, 10, 12

There are 6 items. For the median 6/2 = 3. This is a whole number, so use 3rd and 4th item

(midpoint is 7). For the LQ, 6/4 = 1.5. This rounds up to 2, so use the second item (4).

Hagrid Tip #2: When you have grouped continuous data, and you're finding the

quartiles/percentiles, DO NOT ROUND to find the item number - just keep it as it is. Use

linear interpolation to find an estimate for your quartile/percentile.

Example: Consider our third data set again. We add a cumulative frequency row:

Weights of owls (kg) Frequency Cumulative Frequency

To calculate the Lower Quartile: , so 4th item. The 4th item occurs within the

first 5 items. We can put the cumulative frequency at the start and end of the matching

interval, as well as the item we're interested in. We can also put the class boundaries on the

bottom side of the line.

We're clearly 4/5 of the way along the line, so we go 4/5 of the way along from 3kg to 5kg:

(

)

To calculate the 72th Percentile, :

72% of 16 = 11.52th item. This doesn't occur within the first 5 items but does

occur within the first 13 items, so we know is in the weight interval.

Thus (

) .

Hagrid Tip #3: Be vigilant of gaps in class intervals vs no gaps, and of dark wizards.

Suppose we instead had the following data:

Weights of owls (kg) Frequency Cumulative Frequency

We now have gaps! If you don't adjust the class intervals accordingly (i.e. the class widths would

be 3, 3 and 5), you'll get absolutely no marks for your linear interpolation. On the plus side

however, Voldemort would consider enlisting you as a Death Eater.

For example, to find the median:

so use 6th item.

5kg 6kg 5

13 11.52

?

3kg 5kg

0 5 4

?


(

)

Hagrid Tip #4: Try and memorise the mnemonic for the formula for variance, and how the

formula results from it, rather than memorise the formula itself.

Hagrid Tip #5: Make sure you understand the difference between and ( ) .

Hagrid Tip #6: Use your calculator to check your value of the variance (see Calculator Tips

above).

The mnemonic for variance: "The mean of the squares minus the square of the mean" ("msmsm").

This gives:

1. Ungrouped data:

(

)

(since

)

2. Grouped data:

(

) . Don't get these two mixed up!

You should not however think of these formulae as different. clearly means the same as . And

still means the (estimated sum) of the values (using the midpoints of the class intervals).

Confusingly, when exam questions use a variable for the values, say , but the data is grouped,

still refers to the total of all the values with the frequencies factored in. This is likely to be different

to , because the latter is an estimate of the total using the midpoints of the grouped data,

whereas is the exact total of the values before the data was grouped and information was lost

(see Edexcel Jan 2011 Q5 for example). This just means that if you wanted the mean of and you

were given , then

, and ignore the grouped frequency table you were given.

Hagrid Tip #7: Don't forget to square root when you're finding the standard deviation from the

variance!

Hagrid Tip #8: Check that your standard deviation looks sensible. Standard deviation roughly

means "the average distance from the mean". So if your standard is 10 times too large say, then

you know you've gone wrong.

Coding:

1. However you code your variable (adding, dividing, etc.) you do the same to the mean.

2. Adding or subtracting to your variable doesn't affect the spread (variance/standard

deviation). This intuitively makes sense: were everyone to get exactly 50cm taller by

standing on a chair, the heights are just as spread out.

3. Multiplying or dividing affects the standard deviation in the same way. If you double the

heights, you double the standard deviation. You halve the heights, you halve the standard

deviation.

4. For variance though, we have to square the factor difference. For example, if the values

tripled and hence becomes , then the variance is ( ) , i.e. the variance becomes

9 times larger.

Hagrid Tip #9: Make sure you check whether you're finding the new mean/variance/standard

deviation after coding, or the original mean/variance/standard deviation before coding.

5.5kg 8.5kg

3 7 6

?


Chapter 4 - Data: Location and Spread _

Box Plots

Remember that you need to calculate your outlier boundaries, which are generally 1.5 Interquartile

Ranges above the UQ or below the LQ. You will always be told in the exam question however how

the outlier boundary is defined.

Buckbeak Tip #1: There's two possibilities for the end points of the whiskers when there's

an outlier on that end, and mark schemes accept both: either use the outlier boundary

itself, or the smallest/greatest value which is not an outlier.

Buckbeak Tip #2: You must explicitly show your calculation for the outlier boundaries.

There's marks specifically for this in the mark scheme, and if you display your whiskers

slightly incorrectly, you'll risk losing all marks.

Stem and Leaf Diagrams

You may be asked to calculate the interquartile range. In which case, just remember that you have a

discrete list of items, and hence choose the items to use for the quartiles in the correct way.

When asked to compare the two sets of data in a back-to-back stem and leaf diagram, they're

expecting things like "the boy's scores tend to be higher than the girls".

Histogram

Pretty much all histogram questions boil down to this simple diagram:

i.e. You're identifying the scaling ( ) from area to frequency. At GCSE you could always assume that

, i.e. area is EQUAL to frequency. Identifying may come from either using the total area and

total frequency given (when frequencies of individual intervals are not available) or from the known

frequency and area of a particular bar.

Once is known, you can use it to calculate frequencies for any area.

Use my slides for practice: http://www.drfrostmaths.com/resource.php?id=11371

Buckbeak Tip #3: If you're not given a frequency density scale on the histogram on the -

axis, and only know the total frequency, just add any frequency density scale. If you know

the frequency of a particular bar, it's generally easiest to set the scale such that

.

Buckbeak Tip #4: As before, you must check if the class intervals have gaps! If so, ensure you

use the correct class intervals when calculating frequencies/frequency densities.

Buckbeak Tip #5: When asked to find the mean, median, quartiles or variance of a

histogram, first use the histogram to generate a grouped frequency table. Then use this

table as you usually would to calculate these statistics.

Buckbeak Tip #6: When asked why a histogram is an appropriate means of displaying the

data, the words they're looking for are 'continuous data/variable', and nothing else.

Area Frequency


For questions which ask you the width/height of a bar in a histogram: Find the scaling between the

class width of a bar and its width, and between the frequency density of a bar and its height.

e.g. If a bar with class width 3 and frequency density 10 has ‘drawn’ width 6cm and height 5cm, then

the scale factors are 2 and 0.5 respectively. Then if some other bar has class width 5 and frequency

density 8, then its width and height are 10cm and 4cm.

Skew

Remember that there's 3 ways in which you calculate skew:

1. For histograms or probability distributions, just observe the shape. If the 'tail' is in the

positive direction, you have positive skew. If it's in the negative direction, you have negative

skew.

2. Use the quartiles. If the right box of the (implied) box plot is wider, i.e. ,

then you have positive skew. If the left box is wider, you have negative skew. If they're the

same width, you have no skew.

3. Use the mean and median. The way to remember which order means which type of skew is

to think of salaries: Large salaries in the positive tail drag up the mean but not the median,

hence means we have positive skew.

On the rare occasion, you have both the quartiles and mean available. In which case, choose either

(2) or (3) to find what type of skew you have. Otherwise, the choice should be clear based on the

data available.

Buckbeak Tip #6: For 2 mark questions which ask you to comment on skew, you get one

mark for saying 'negative/positive/no skew', and the other mark for given a valid reason (e.g.

).

Chapter 5 - Probability _

If and are independent, then does not affect and vice-versa.

If and are mutually exclusive, then and can’t happen at the same time.

These are completely separate things – one is not the opposite of the other!

Laws

If A and B are independent:

( | ) ( )

(as the probability of A is not affected

by B)

( ) ( ) ( )

(If you're asked to show that two

events are independent, then show

this equality holds)

If A and B are mutually exclusive:

( )

( ) ( ) ( )

In general:

( | ) ( )

( )

(I remember this by ‘the intersection divided

by the thing I’m conditioning on’)

(Notice that if A and B are independent, then

the RHS simplifies to ( ))

Addition Rule: ( ) ( )

( ) ( ) (Remember this by thinking about two

overlapping circles – we need to subtract the

overlap)

(Notice that if A and B are mutually exclusive,

( ) , so we get our earlier formula)

(If A and B are independent, this reduces to

( ) ( ) ( ) ( ) ( )


McGonagall Tip #1: Mutually exclusive events are indicated by separated non-overlapping

circles in a Venn Diagram. Independence does not affect the Venn Diagram.

McGonagall Tip #2: If you’re not told two events are mutual exclusive, then for the purposes

of the Venn Diagram, you have to assume that they are not mutually exclusive, i.e. they

overlap.

McGonagall Tip #3: You can often determine probabilities by constructing a Venn Diagram

and filling in the missing probability in regions by simple adding/subtracting. Other times,

this approach doesn’t work.

McGonagall Tip #4: You can treat probabilities algebraically.

e.g. ( ) ( ) ( )

McGonagall Tip #5: If you’re told that A and B are independent, immediately write out that

( ) ( ) ( ) using whatever probabilities you’re given. Same for mutual

exclusivity. It’ll help you visualise the probabilities you have available to determine others

you don’t know.

McGonagall Tip #6: Do you have a mixture of ( ), ( ) and ( ) (or ( ))? You

should write out the Addition Rule and see if it helps.

McGonagall Tip #7: Note that given some event, the probabilities add up to 1. So:

( | ) ( | ) and ( | ) ( | )

Some people incorrectly assume:

( | ) ( | )

McGonagall Tip #8: As per GCSE, a suitable tree diagram can work wonders. Remember that

the second level of branching and onwards are conditional probabilities.

Dealing with more complicated tree questions

Suppose you have a tree like the one below:

Then how would we calculate the following?


( ) As per GCSE, we just find all the paths which match this description (i.e. where is true) and add the probabilities of each path:

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) Note that we need not even consider the event because it occurs after :

( ) ( ) ( )

( | ) Seeing the conditional probability, you should immediately go for your formula for conditional probability!

( | ) ( )

( )

( ) ( )

( ) ( )

Chapter 6 - Correlation _

Recall that is 'perfect positive correlation', is 'no correlation' and is 'perfect

negative correlation'. Anything below -0.7 or above 0.7 is considered to be strong correlation.

I'm going to presume here you can plug values into your , , and formulae. But things I

see go wrong:

Lupin Tip #1: In ( )

, I've seen people forget to square the , or mix the

formula up with the one for variable, and do (

) . This formula is clearly

wrong because the in the denominator gets squared when it shouldn't.

Lupin Tip #2: You can use your calculator to directly calculator if you're given the original

data (see the beginning of this guide). Make sure however you still show your calculations

for and so on for the purposes of evidencing working. However, generally you're

generally provided with certain sums in the exam to save you time, so you may not be able

to enter the original data directly.

Here are some potential 'explaining' questions you might encounter:

Lupin Tip #3: If you're asked whether your correlation coefficient supports some assertion,

just comment on whether your value is close to -1, 0 or 1. If someone is claiming that house

prices falls with distance from central London, then their assertion is justified if you have a

correlation coefficient close to -1 (i.e. negative correlation).

Lupin Tip #4: If you're asked to give an 'interpretation' of your correlation coefficient, this

doesn't mean to say whether it's negative or positive, but to say what it actually means in

words. e.g. "Higher towns have lower temperature/temperature decreases as height

increases". i.e. You're asked to state to what happens to one variable as the other increases.


Coding

Completely unaffected by any multiplications, divisions, additions or subtractions in the coding.

Since , is affected by coding in the same way as variance. So if the value is doubled in coding, becomes times bigger. The same applies to .

As above, addition and subtraction in the coding has no effect. If is scaled by a factor of and by a factor of , then is

scaled by a factor of . e.g. If and and

, then .

Chapter 7 - Regression _

Regression in general means to find the parameters of a model which best explains the data. In the

case of linear regression, the model is a straight line, and the parameters are its y-intercept and

gradient, which are set to as to minimises the total (squared) error between the predicted y-value on

the line and y-value on each data value.

The formulae for the equation of the least squares regression line are given in the formula booklet,

but they're worth memorises:

. Notice that unlike , we put the y-intercept term first on the RHS.

This is so that when we extend say to quadratic regression, i.e. fitting a line

, still means the y-intercept.

. I remember this by the fact that 'xy' are the sex chromosomes for a man, and 'xx'

for a woman, and the man comes first (a little bit misogynistic I know, but easy to

remember!)

. This is easily remember by just rearranging above to make the

subject, and then replacing with and with . This suggests that the point ( ) is on

your regression line. Remember that just means the mean of and is calculated using

.

Snape Tip #1: If asked to interpret your gradient, say something like "As [my x variable] increasing by

1, the [your y variable] increases/decreases by [the gradient]", e.g. "as the height increases by 1m,

the temperature decreases by 3 degrees".

Snape Tip #2: It's good to be clear from the outset what's your explanatory variable and what's your

independent variable. This is crucial when you calculator your gradient, since if you get your

variables mucked up, you might do

by mistake. These will be rarely labelled using the

variable letters and , so ensure you work out which is which. If for example your had the water

depth and the pressure , clearly depends on , so is your (explanatory) variable and your

(dependent) variable. Don't write (since and don't exist in this context), use instead.

Snape Tip #3: Remember that for coding, your just replace your variables using the expressions

given. e.g. If you have the line and you used the coding , , then just

substitute: ( ) and simplify. Piece of cake.


Chapter 6-7 - Interpretation Questions for Correlation/Regression _

Interpretation questions are a bit of a pain if you don't know the examiner is looking for, and often

worth a lot of marks (anything up to 4!). Here's everything you need to know:

Question Answer format Comments

Interpret r "The graph show a ___ correlation. As ____ increases, ____ increases/decreases."

The key here is saying how the dependent value changes as the explanatory variable increases. The question asks you to 'interpret', so it's key here you describe what it explains using NON-STATISTICAL LANGUAGE. Saying "Positive correlation" alone would be DESCRIBING the correlation.

Explain why this diagram would support the fitting of a regression line of y onto x

"The variables have a linear relationship, i.e. the points are close to the implied straight line of best fit."

Interpret the gradient/slope of the line/interpret b

As (x) increases by 1, (y) increases/decreases by ___.

Again, the key word here is 'interpret'. Example "As the height increases by 1m, the temperature decreases by

Interpret the y-intercept/interpret a

The value (y) takes when (x) is 0.

e.g. "A score of 55 would be obtained if no hours of revision are done".

Extrapolation "Reliable [1 mark] because the value is within the range of the data [1 mark]." "Reliable because the value is close to the range of the data". "Unreliable because the value is outside the range of the data, i.e. we are extrapolating."

The 'tricksey' case here is when the value is just outside the data range. The examiner is expecting you to state the your regression equation is still reliable.

Which is the explanatory variable? Explain your answer.

"(x) is the explanatory variable because (x) influences (y)".

Explain method of least squares

"We minimise the square of the residuals" (draw a diagram)

You may wish to go back to your textbook to understand what's going on here. But the principle is that we're finding the error between each data value and the predicted value (based on the model), squaring the error (to ensure the error is positive) then adding them up. The gradient and y-intercept of the straight line is set to minimise this total error.


Chapter 8 - Discrete Random Variables _

A discrete random variable has two ingredients:

1. A set of outcomes.

2. A probability distribution mapping outcomes to probability. This has two flavours:

a. A 'probability distribution' is represented either as a table or as a graph.

b. A 'probability function' allows you to have a more complex expression that

calculates the probability, as opposed to a probability distribution where each

outcome and its probability are explicitly stated. These might be specified as a

'piecewise function', where different functions are used for different ranges of

outcomes, e.g:

( ) { ( )

( ) is the longhand for writing a probability. It means "the probability that a random variable

has the value . ( ) is the shorthand. Note the use of lowercase .

A cumulative distribution function meanwhile gives the probability up to a certain value, i.e.

( ) ( )

Finding the value of a missing variable using a probability function or cumulative distribution

function

( ) { ( )

Probabilities must add up to 1. So just plug in

each value of into your probability function:

So

( ) { ( )

It's clear ( ) , because it's certain that we'll

have an outcome up to 4.

This gives us , so

Finding a probability distribution from a cumulative distribution function

Since the cumulative distribution function is the running total of the probability, finding the

difference tells us what we added to the running total each time.

Expected Value

Expected value just means the 'mean' of the random variable. So for a fair die, ( ) because

we expect to see an outcome of 3.5 on average. For this reason ( ) is sometimes represented as

. We just sum the product of each outcome with its probability:

( ) ( )


Finding the value of missing variables using a probability distribution and a given expected value

We form two simultaneous equations by:

1. Noting that the probabilities of a probability distribution add up to 1.

2. Using the provided expected value.

e.g. Given that ( ) and that has the probability distribution below, determine and .

( )

Then just simplify and solve.

Variance

The mnemonic "the mean of the square minus the square of the mean" still applies!

( ) ( ) ( )

If you can't distinguish between ( ) and ( ) , then it's likely you're one of those people who

mix up and ( ) . ( ) means you find the expected value as before, except you replace each

outcome with itself squared - the probability is always unaffected, i.e. ( ) ( ).

Coding

The same rules apply as before. For expected value, whatever we did to the random variable, we do

to the expected value. Adding/subtracting has no effect on variance, and when we multiply or

divide, we square this value.

Examples: ( ) ( ) ( ) ( )

Dealing with inequalities

Have to deal with say ( )? Just solve the inequality first:

( )

( )

Thus we’d add the probabilities for .


Chapter 9 - Normal Distribution _

A normal distribution is perfect for data which is symmetrically distributed about some mean and

the probability tails off as we get further from the mean.

Some preamble

z-values and -tables

The value means the number of standard deviations above the mean. For IQ for example, where

( ), and IQ of 130 corresponds with a -value of 2, and an IQ of 85 with a -value of -1.

This makes sense: is how far about the mean a value is, so dividing it by tells us how many

standard deviations above the mean we are.

If is the original random variable, converting it to a new random variable using the formula

above is known as standardisation. is a normally distributed variable where ( ), known as

the standard normal distribution. The reason and is because then if the value is say 3,

then this IS the number of standard deviations above the mean, because 3 is 3 lots of 1 above 0.

Thus in converting to , each value of the random variable now represents the number of

standard deviations above the mean. So an IQ of 130 gets converted to 2, and so on.

A -table allows us to find out the probability of being up to a certain value, i.e. ( ), where

is a specific value of . Looking up in the table would determine the probability of being up

to 3 standard deviations above the mean; in the case of IQ, it would tell us the probability of having

an IQ less than 145.

Manipulating z-probabilities

Just think about the graph of a normal distribution when manipulating probabilities. ( )

represents the probability of being less than two standard deviations below the mean, e.g. having an

IQ less than 70. Due to symmetry, this is the same probability as being at least 2 standard deviations

above the mean, e.g. having an IQ greater than 130. i.e. ( ) ( ).

Similarly, since the probability of having an IQ above 130 is one minus probability of having an IQ

below 130, we have that ( ) ( ).

In order to be able to use the z-table, we (a) need our z-value to be positive and (b) the inequality to

be <. The quick not-having-to-think way is this: If we change either the direction of the inequality

or we change the sign, we do 1 minus the probability. If we do both, these ‘cancel each other out’

and thus has no effect on the probability:

( ) > is wrong way and -3 is negative, so ( )

( ) > is wrong way, so ( )

( ) < is right but -0.5 is negative, so ( )


Finding a probability

"Let represent the IQ of a randomly chosen person, where ( ). Find the probability

that a randomly chosen person has an IQ below 96".

Step 1: Express your information as a probability. ( )

Step 2: Standardise, i.e. convert your X value to a Z value, using the formula (or common sense). Ensure you get the sign of z correct.

(

) ( )

Step 3: Manipulate your z probability so that the z value is positive and you're using <.

( ) ( )

Step 4: Look up in z table.

Step 5: Check that your answer looks sensible. (Yes, the answer looks sensible because we expected slightly less than half the population)

Finding a value corresponding to a probability

Example 1: "Find the lower quartile for IQ, given that ( ) again."

Step 1: As before, express your information as a probability.

( )

Step 2: Standardise. We don't like the probability of 0.25 because it won't be in our z-table (it's less than 0.5). So we have to do 1 minus it to get 0.75. Thus we have to either change < or > (which we don’t want to do) or turn to .

( ) ( )

Step 3: Ordinarily, we'd look up the closest z-value that gives that probability, which turns out to be 0.67. For certain 'nice' probabilities (e.g. 2.5%, 5%, 10%), we MUST look in the second z-table, otherwise you lose marks. However, in this case, 0.75 is not there, so use first table.

This makes sense as we expect our value to be negative as it's below the mean.

Step 4: Convert back into an X value by using the formula for z:

I personalise find it easier to write ( ) as it expresses the fact that is 0.67 standard deviations below the mean (i.e. 0.67 lots of 15 below 100). However, since the form at the top of this box is what's listed for the method mark, it's probably safer to do it that way.

Example 2: "Find the IQ for which 10% of the population exceeds, given that ( ) again."

Step 1: As before, express your information as a probability. Clearly if 10% of people have an IQ higher than this value, then there's 90% below (recall that we always want the probability of being below some value on the right half of the graph in order to use the z-table).

( )

Step 2: Standardise. No manipulation of our z probability required as we have < and a probability greater than 0.5.

( )


Step 3: This time we can use the second z-table because 0.1 is in it (note that confusingly, the second table gives us the probability of being ABOVE some z-value, i.e. ( )).

Step 4: As before, use the formula for z to convert back into an IQ.

Finding the missing mean or standard deviation

You're highly advised to do lots of practice questions on this until it's second nature!

Example: "Only 10% of maths teachers live more than 80 years. Triple that number live less than 75

years. Given that life expectancy of maths teachers is normally distributed, calculate the standard

deviation and mean life expectancy."

Step 1: As before, express your information in terms of probabilistic statements. Notice we've turned the "10%

above" into "90% below".

( ) ( )

Step 2: Deal with first. ( ) ( )

Step 3: Deal with second. ( ) ( ) ( )

Step 4: We have two simultaneous equations, so solve!

Subtracting the two equations:

Dealing with probability ranges

These questions come in two flavours. The strategy is to go from ( ) to the form

( ) so you can solve in the conventional manner. Suppose that ( ), i.e. IQ:

1. The probability is between the mean and some value.

“If ( ) , then find the value of .”

Either note that ( ) ( ) ( ) ( ) , or by

drawing a suitable graph, we can see that the probability up to is the probability up to 100

(i.e. 0.5) and the probability between 100 and . Thus:

( )

and proceed as normal.

2. The probability is symmetrical about the mean.

“If ( ) . Find .”

Again using a graph, we see that if the middle section is 0.3, then by symmetry, each tail is

0.35. Thus the probability up to is

( )


Dealing with ‘redundant events’

By redundant event I mean one which is implied by another. For example, if I said “find the

probability someone’s age is above 30 and above 40”, this is equivalent to just saying “above 40”.

This is particularly vital for the increasingly common conditional probability questions in Normal

Distribution questions:

( | ) ( )

( )

( )

( )

I find your S1 efforts most... a- -sing.

Harry Potter stats puns (c) Lord Voldermort.

Copyright infringement punishable by Cruciatus.

Documents

Harry Potter Guide to s 1