BSNS112 Master

Quantitative Data

Discrete and Continuous

Discrete Data

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

Continuous Data

Measured infinitely such as age, height, time

Qualitative Data

Categorical, ordinal and nominal

Ordinal Data

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

Nominal Data

Does not convey ranking such as ethnicity, gender

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

Discrete

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

Ordinal - conveys a ranking

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

Nominal, conveys no ranking

What type of data is time spent at the market?

Discrete - measures time which is a specific value

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)

Cross tabulation

Compares categorical with categorical

scatter plot

Compares numerical with numerical

frequency table

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

Stacked / clustered bar chart

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

Relative frequency histogram

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

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

Mean

simple average

median

middle most value (when ranked from ascending to descending)

mode

most frequent

trimmed mean

without most extreme 5%

Range

maximum - minimum

interquartle range

75th percentile minus 25th percentile

variance

represents spread of data around the mean. Standard deviation squared

standard deviation

square root of variance, higher spread means more spread

co-efficient of variation

compares different groups with different magnitudes to compare variability

skewness

positive = right negative = left

significantly skewed

data is skewed more than twice its standard error

mode

median

Kurtosis

measures the extent to which observations cluster around the central point

What is it called when the kurtosis statistic is zero?

normal distribution

data clusters close to centre: positive or negative kurtosis?

positive

data clusters further from centre: positive or negative kurtosis?

negative

co-variance

measures co-movement between 2 variables

correlation of co-efficient

measures the linear relationship between 2 variables

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

scatterplot, as it measures numerical by numerical

population

whole collection under analysis

sample

a portion of the population

parameter

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

statistic

summary measure computed to describe a characteristic of a sample

primary data

collected yourself

secondary data

taken from another source

observational data

you observe and record

experimental data

data you've obtained through experiments

simple random sampling

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

systematic random sampling

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

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

cluster sampling

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

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

Non sampling errors

human errors

coverage errors

when the sample has targeted the wrong subjects

non-response error

when subject chooses to not respond, impacting the data

measurement error

caused by bad question and misunderstanding

margin of error

quantified measure of sampling error

probability

how likely an event is to occur

how is probability written

P(event)

What is U in probability

union - probability of one event occurring over another

what is 'n' in probability

intersection - probability that both events occur together

collectively exhaustive

when the outcomes given are the only possible outcomes

complement

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

A Priori Classical

when you already know the probability exists through information

Empirical (relative frequency)

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

Subjective

when the probability is based in your opinion

Conditional Probability

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

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"

how is conditional probability calculated?

P(a n b) / P(b)

Marginal probability

total probability of a row or column

Probability independence

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

When does co-variance = 0?

when variables are independent

Random Variables

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

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.

Expected Value defined

the value we expect based on the probabilities that exist.

Expected Value formula

E = ∑ [x • P(x)]

Variance

measures data spread around the mean

Variance formula

V(X) = ∑ [p(xi) + (xi-M)^2]

Binomial Distribution

discrete probability distribution with 4 characteristics

what are the 4 binomial characteristics

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

2. fixed number of trials

3. probability of success remains the same for every trial

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

Discrete Random Variables

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

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

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

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)

4 characteristics of Poisson

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

2. Probability is the same for all equal sized intervals

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

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

Empirical Rule

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

normal distribution

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

Standardized Z-Distribution

mean = 0 standard deviation = 1

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

Graphs to show normally distributed data

1. histogram

2. box plot

3. stem & leaf

4. qq pp plot

What does a sample statistic do

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

A quantitative estimate involves

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

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

A qualitative estimate involves

a proportion "what proportion of the population is from christchurch

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

Point Estimates

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

how to calculate confidence intervals

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

standard error

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

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

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

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

What are the Z values

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