IB SL Maths AA

formula for median

(n+1)/2 (gives position of the median, not the value!)

(𝒙𝑦)³ = ?

(𝒙𝑦)³ = 𝒙³𝑦³

The sum of an infinite series can only be found when...

-1 < r < 1, r ≠ 0

Pascal's Triangle Row 10

1 10 45 120 210 252 210 120 45 10 1

Pascal's Triangle Row 11

1 11 55 165 330 462 462 330 165 55 11 1

Pascal's Triangle Row 5

1 5 10 10 5 1

Pascal's Triangle Row 6

1 6 15 20 15 6 1

Pascal's Triangle Row 7

1 7 21 35 35 21 7 1

Pascal's Triangle Row 8

1 8 28 56 70 56 28 8 1

Pascal's Triangle Row 9

1 9 36 84 126 126 84 36 9 1

π radians

180 degrees

360° equals

2π radians

Circumference of a circle

2πr

π/4 radians

45 degrees

1 radian equals

57.3°

π/3 radians

60 degrees

π/2 radians

90 degrees

90th percentile

90% of the data

census

A complete enumeration of a population.

Correlation

A measure of the extent to which two factors vary together, and thus of how well either factor predicts the other.

cluster sampling

A probability sampling technique in which clusters of participants within the population of interest are selected at random, followed by data collection from all individuals in each cluster.

systematic sampling

A procedure in which the selected sampling units are spaced regularly throughout the population; that is, every n'th unit is selected.

quota sampling

A sampling technique in which the population is divided into groups and a sample size is taken from each stratum which is in proportion to the size of the stratum (e.g. In a high school of 1000 students where 60% of the students are female and 40% are male, your sample should should also be 60% female and 40% male)

stratified sampling

A type of sampling in which the population is divided into groups with a common attribute ('strata') and a random sample is chosen within each 'stratum'. They are then put together to form a sample

Give an example of an infinite geometric series where it is impossible to find the sum

Any infinite geometric sequence where r < -1, r > 1

Give an example of infinite geometric series with a finite sum

Any infinite geometric sequence where │r│< 1, r ≠ 1

disadvantages of median

Cannot estimate population parameters and can be unrepresentative, not as sensitive as the mean

multimodal

Describes a graph of quantitative data with more than two clear peaks.

Compound Interest Formula

FV = PV(1+r/100k)^kn fv = future value pv = present value n = number of years k = number of compounding periods per year r = nominal annual interest rate A=P(1+(r/n))^nt p = principle amount r = rate n = times componded t = years

Interpolation

From within the given domain

Simple Interest Formula

I = prt Interest = (amount $)(rate as decimal)(time in yrs) A = P(1+rt) P = Principal (Initial Amount)

disadvantages of mean

Is sensitive to every data value, one extreme value can affect it dramatically; is not a resistant measure of center

advantages of median

It is easy to calculate. It is not heavily affected by extreme values/anomalies or lack of symmetry.

disadvantages of mode

It isn't always present or there may be more than one. It does not use all the data.

Pearson product-moment correlation coefficient

Measure of the correlation strength between two variables x and y.

outlier

More than 1.5xIQR above Q3 More than 1.5xIQR below Q1

When a power is raised to a power... (𝒙3)³ = ?

Multiply them (𝒙³)³ = 𝒙⁹

discrete data

Numerical data values that can be COUNTED

continuous data

Numerical data values that can be MEASURED

Extrapolation

Outside of the given domain

Lower boundary

Q1-1.5(IQR)

Upper boundary

Q3 + 1.5(IQR)

independent variable (x)

The experimental factor that is manipulated; the variable whose effect is being studied.

dependent variable (y)

The outcome factor; the variable that may change in response to manipulations of the independent variable.

advantages of mean

Uses all the data, usually most representative

x ≥ -4

[-4, ∞[ [-4, ∞)

6 ≤ x ≤ 9

[6, 9]

-1 < x < 5

]-1, 5[ (-1, 5)

x > -4

]-4, ∞[ (4, ∞)

x < 3

]-∞, 3[ (-∞,3)

x ≤ 3

]-∞, 3[ (-∞,3]

normal distribution

a bell-shaped curve, describing the spread of a characteristic throughout a population

negatively skewed

a distribution of scores in which scores are concentrated in the high end of the distribution

positively skewed

a distribution of scores in which scores are concentrated in the low end of the distribution

Bivariate

a set of data that has two variables

A sample is...

a subset of the population that will give you information about the population as a whole

discrete data is represented on

bar chart/column graph

convenience sampling

choosing individuals who are easiest to reach

bimodal

distributions with two modes

Just because two variables are correlated...

does not necessarily mean one causes the other.

Anything to the power of 0

equals 1

The population consists of...

every member in the group that you want to find out about

simple random sampling

every member of the population has an equal probability of being selected for the sample

A negative number taken to an odd power...

gives a negative result

A negative number taken to an even power...

gives a positive result

unimodal

having one mode; this is a useful term for describing the shape of a histogram when it's generally mound-shaped

continuous data is represented on

histogram

Anything to the power of 1

is itself

regression line

is the line of best fit

-0.87 < r ≤ -0.5

moderate negative correlation

0.5 ≤ r < 0.87

moderate positive correlation

Median (cumulative frequency)

n/2 (gives position of median, not value)

-0.1 < r ≤ 0

no correlation

0 ≤ r < 0.1

no correlation

r=-1

perfect negative correlation

r=1

perfect positive correlation

standard deviation

provides a measure of the standard, or average, distance from the mean spread of the data (the square root of the variance) s, sd, σ

linear correlation

relation between two variables that shows up on a scatter diagram as the dots roughly following a straight line

variance

standard deviation squared

-0.95 < r ≤ -0.87

strong negative correlation

0.87 ≤ r < 0.95

strong positive correlation

Lower Quartile (Q1)

the median of the lower half of the data (n+1/4) (gives position, not value) 25th percentile

Upper Quartile (Q3)

the median of the upper half of the data 3(n+1/4) (gives position, not value) 75th percentile

Interquartile Range (IQR)

the middle 50% of the data Q3-Q1

Data is sufficient if...

there is enough data available to support your conclusions

-1 < r ≤ -0.95

very strong negative correlation

0.95 ≤ r < 1

very strong positive correlation

-0.5 < r ≤ -0.1

weak negative correlation

0.1 ≤ r < 0.5

weak positive correlation

(𝒙/𝑦)⁴ = ?

x⁴/y⁴

Data is reliable if...

you can repeat the data collection process and obtain similar results

advantages of mode

•it is the only one that can be used with nominal data •unaffected by extreme values •more useful for discrete data

Natural Numbers

ℕ Counting nominal numbers (no negatives): 0, 1, 2, 3, 4..

Rational Numbers

ℚ Any number that can be expressed as a fraction

Irrational Numbers

ℚ¹ π, e, √2, √3

Real Numbers

ℝ

Integer

ℤ