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Quantitative Data

Discrete and Continuous

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Discrete Data

Must be measured in specific order / values, such as number of students in a class

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Continuous Data

Measured infinitely such as age, height, time

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Qualitative Data

Categorical, ordinal and nominal

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Ordinal Data

Places in order and conveys a ranking such as clothing sizes (small, medium large)

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Nominal Data

Does not convey ranking such as ethnicity, gender

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What type of data is the number of cars a family owns?

Discrete

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What type of data is the type of accommodation (such as budget, tourist, superior)

Ordinal - conveys a ranking

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What type of data is favourite fruit preference at the market?

Nominal, conveys no ranking

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10

What type of data is time spent at the market?

Discrete - measures time which is a specific value

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Weekly household spending is divided into these groups: less than $50, $50-$100, $150-$200. What type of variable is this?

Categorical & Ordinal (defines categories and placed in order to convey a ranking)

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Cross tabulation

Compares categorical with categorical

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scatter plot

Compares numerical with numerical

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frequency table

analyses 1 categorical variable. E.g. the fave stall of people at the market

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Stacked / clustered bar chart

compares categorical with categorical e.g. proportion of M/F choosing fave stall

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Relative frequency histogram

compares categorical with numerical (e.g. market spend of various occupational groups)

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If the 2 variables are, "favourite stall" and "if visitors are regular or not", ac ross tabulation should be used because,

both variables are categorical and define a particular category

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Mean

simple average

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median

middle most value (when ranked from ascending to descending)

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mode

most frequent

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trimmed mean

without most extreme 5%

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Range

maximum - minimum

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interquartle range

75th percentile minus 25th percentile

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variance

represents spread of data around the mean. Standard deviation squared

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standard deviation

square root of variance, higher spread means more spread

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co-efficient of variation

compares different groups with different magnitudes to compare variability

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skewness

positive = right negative = left

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significantly skewed

data is skewed more than twice its standard error

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mode

median

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Kurtosis

measures the extent to which observations cluster around the central point

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What is it called when the kurtosis statistic is zero?

normal distribution

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data clusters close to centre: positive or negative kurtosis?

positive

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data clusters further from centre: positive or negative kurtosis?

negative

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co-variance

measures co-movement between 2 variables

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correlation of co-efficient

measures the linear relationship between 2 variables

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What graph would measure the following: comparing time spent at the market average income

scatterplot, as it measures numerical by numerical

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population

whole collection under analysis

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sample

a portion of the population

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parameter

summary measure describing a characteristic of the data, a type of rule or limit

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statistic

summary measure computed to describe a characteristic of a sample

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primary data

collected yourself

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secondary data

taken from another source

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observational data

you observe and record

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experimental data

data you've obtained through experiments

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simple random sampling

everyone is equally likely to get chosen from the population. E.g. randomly picking a certain number of students

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systematic random sampling

having a system when randomly selecting sample. E.g. randomly selecting a sample then every K'th sample thereafter

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Stratified random sampling

dividing populaiton into homogenous groups (similar characteristics) then taking random sample, e.g. dividing students by which degree they take then taking random sample

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cluster sampling

dividing population into several clusters that aren't homogenous but are each representative of the population then taking a random sample

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You want to sample residential halls but worry that a random sample wont include the small halls. Which sampling method should you use?

Stratified random sampling

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Non sampling errors

human errors

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coverage errors

when the sample has targeted the wrong subjects

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non-response error

when subject chooses to not respond, impacting the data

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measurement error

caused by bad question and misunderstanding

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margin of error

quantified measure of sampling error

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probability

how likely an event is to occur

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how is probability written

P(event)

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What is U in probability

union - probability of one event occurring over another

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what is 'n' in probability

intersection - probability that both events occur together

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collectively exhaustive

when the outcomes given are the only possible outcomes

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complement

2 events complement each other if their probabilities add to 1. E.G. P(a) + P(b) =1

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A Priori Classical

when you already know the probability exists through information

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Empirical (relative frequency)

when you choose to work out the probability through experiments rather than information

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Subjective

when the probability is based in your opinion

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Conditional Probability

the probability of an event occurring given that another event has already occurred.

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How is conditional probability written

P(A I B) e.g. P(Student I Female) "what is the probability that it is a student and they're female"

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how is conditional probability calculated?

P(a n b) / P(b)

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Marginal probability

total probability of a row or column

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Probability independence

when the probability of one event does not influence the probability of another event occurring

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When does co-variance = 0?

when variables are independent

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Random Variables

variables with multiple possible values and an associated probability of getting each variable

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Discrete Random Variables

can only take on a finite number of variables, e.g. the number of 6's rolled on a dice over 2 rolls: there can only be either 0 sixes, 1 six, or 2 sixes.

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Expected Value defined

the value we expect based on the probabilities that exist.

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Expected Value formula

E = ∑ [x • P(x)]

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Variance

measures data spread around the mean

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Variance formula

V(X) = ∑ [p(xi) + (xi-M)^2]

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Binomial Distribution

discrete probability distribution with 4 characteristics

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what are the 4 binomial characteristics

has to be 2 outcomes to every trial (success or fail)

fixed number of trials

probability of success remains the same for every trial

trials are independent, where the outcomes don't affect each other).

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Discrete Random Variables

Cannot be divided, whole numbers, e.g. number of phone calls in a day, number of visitors

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Expected Value

what we expect based on previous data. Formula: E(x) = (0 x 0.25) + (1 x 0.5) + (2 x 0.25) = 1

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Variance

spread of the data. Formula is similar to expected value: V(x) = ((0² x 0.25) + (1² x 0.5) + (2² x 0.25))-1²

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Poisson Probability Distribution

A discrete probability distribution used to find probabilities of the number of times a certain event occurs in a specified time interval (no fixed number of trials)

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4 characteristics of Poisson

number of successes in trial is independent of number of successes in any other interval

Probability is the same for all equal sized intervals

probability of success in a trial is proportional to the size of the interval

probability of more than one success in an interval approaches zero as it becomes smaller

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Empirical Rule

68% = 3 standard dev 95% = 2 standard dev 100% = 1 standard dev

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normal distribution

A function that represents the distribution of variables as a symmetrical bell-shaped graph.

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Standardized Z-Distribution

mean = 0 standard deviation = 1

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How to recognise if data is normally distributed

graph is mound shaped and symmetrical

mean = median

empirical rule applies (68=3, 95=2, 100=3)

skewness & kurtosis close to 0

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Graphs to show normally distributed data

histogram

box plot

stem & leaf

qq pp plot

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What does a sample statistic do

makes an inference on a population parameter if you cant sample an entire population.

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90

A quantitative estimate involves

a mean "what is the mean grade of the students"

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what are x̅ and μ

x̅ represents the mean in a sample statistic, and μ is the same as x̅, but it represents the whole (parameter) population

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A qualitative estimate involves

a proportion "what proportion of the population is from christchurch

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Interval Estimates

estimations of a range of values of a population parameter. E.g. we expect μ to fall within $75-$100, or, we expect P to fall within 0.25-0.50

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Point Estimates

estimates an exact value of a parameter using a single value. Unlikely to estimate correctly so use interval estimate instead

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how to calculate confidence intervals

point estimate plus or minus margin of error (confidence level x standard error)

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standard error

is the standard deviation of sample mean/proportion and represents the sample mean/proportions accuracy

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when would you use the z distribution when trying to estimate a confidence interval

when the population standard deviation is known

the sample is normally distributed or, sample is large

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When would you use the t distribution when trying to estimate a confidence interval

population standard deviation is unknown

sample is normally distributed or, is large

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99

when would you use the Z distribution when trying to estimate a confidence interval

for proportions as you'll always know the population ST.D

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100

What are the Z values

99% = 2.576 95% = 1.96 90% = 1.645

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