Danny AP Statistics

All items or individuals of interest

Population

Number of items or individuals in the population:

Population size

Subset of population from which data are collected:

Sample

Number of items or individuals in the sample:

Sample size

Symbol for Population size:

N

Symbol for sample size:

n

A study in which data are collected from a sample to answer an investigative question about a larger population:

Statistical study

An item or individual from which data is collected:

Observational unit

A characteristic or attribute that varies between observational units:

Variable

What does a categorical variable take on?

Values that are category names or group labels

What are categorical variables also called?

Qualitative variables

Do categorical variables have an order?

No

Examples of categorical variables:

Left right hand

What does a quantitative variable take on?

Numerical value

Do quantitative variables have an order?

Yes

Types of quantitative variables:

Continuous and discrete

What can discrete variables take on?

Distinct set of values or some specific values

Example of discrete data:

Only whole numbers

What can continuous variables take on?

Any value within an interval

A numerical attribute among the whole population:

Parameter

A numerical attribute among the measured sample:

Statistic

What does a frequency table show?

The frequencies in each category of a categorical variable

What are frequencies also called?

Counts or number of repeatedness

What does a relative frequency table show?

Relative frequencies in each category of a categorical variable

What are frequencies also called?

Proportions or percentage

Are bars of bar charts separated?

Yes

What are bar charts used for?

Categorical variables or discrete quantitative variables

Advantages of dotplots:

Can see every data point on full scale

Disadvantages of dotplots:

Depending on scale, exact value will disappear

Only what of values is shown for dotplots?

Relative location

What does the accuracy of dotplots depend on?

Tools

Advantage of stem and leaf plot:

Can see values of every data point

Disadvantage of stem and leaf plots:

Overwhelming for large datasets

Do bars attach with each other for histograms?

Yes

What type of data are histograms used for?

Continuous quantitative

Advantage of using histogram:

Good for summarizing large datasets

Disadvantage of using histogram:

Can’t see individual data values

Acronym for describing a distribution:

CUSS with context

What does CUSS with context stand for?

Center, unusual features, shape, spread

What are unusual features of a distribution also called?

Outliers or extremes

What must exist for there to be unusual features in a distribution?

Gaps and outliers

Words to describe shape of distribution:

Uniform, unimodal, bimodal, left skew, symmetric, right skew

Where is the tail of a left skew?

Left

What is a symmetric distribution also called?

Bell shaped curve

Where is the tail of right skew?

Right

What are the four stages involved in statistics?

Collection of data, organization of data, presentation of data, analysis of data

Useful tools for analysis of data:

Measures of central tendency and measures of dispersion

Example of grouped data:

Ages 1-10, 11-20, 21-30

How to calculate arithmetic mean with ungrouped data?

Add up all values and divide by number of values

How to calculate arithmetic mean for grouped data?

(x1f1 + x2f2 + … + xnfn) / (f1 + f2 + … + fn) where x are class marks and f are the frequencies of class marks

How to calculate weighted mean for grouped data?

(x1w1 + x2w2 + … + xnwn) / (w1 + w2 + … + wn) where x are the data and w are the weights of the data

What does weighted mean of grouped data use as weight?

Frequency of each group

Datum with highest frequency:

Mode

Can there be no mode / modal class?

Yes

Can there be more than one mode / modal class in a data set?

Yes

Class interval with the highest frequency:

Modal class

How to find median?

Arrange data in ascending order of magnitude, if N is odd number, median is ((N+1)/2)th datum. If N is even number, median is 1/2((N/2)th datum + (N/2 + 1)th datum)

What can the median of a set of grouped data be found from?

50th percentile in cumulative frequency polygon or curve

What happens to mean, median, and mode when constant C is added to each value?

Increase by c

What happens to mean, median, and mode when each value is multiplied by constant c?

Multiplied by c

What happens to mean, median, and mode when value 0 is added to a set of positive values?

Mean decreases, median decreases/remains the same, mode unchanged

What happens to the mean, median, and mode if a value is removed?

Mean increases/decreases is value is smaller than / greater than arithmetic mean, Median will increases/decrease if value is smaller than / greater than median or remains unchanged, mode unchanged if value is not mode

If an item which is equal to the mean is added / removed, the mean:

Remains unchanged

How to calculate range for ungrouped data:

Largest datum - smallest datum

How to calculate range for grouped data:

Highest class boundary - lowest class boundary

How to calculate inter-quartile range

Upper quartile (Q3) - lower quartile (Q1)

How to calculate standard deviation of population:

How far on average is each datum away from the mean

Standard deviation

How to calculate variance?

(standard deviation)^2

Advantages of range:

Simple, suitable if max and min data are most important

Disadvantages of range:

Only 2 data used, extreme values make range large

Advantages of interquartile range:

Not affected by extreme values

Disadvantages of interquartile range:

Data has to be sorted first, only part of data used

Advantages of standard deviation:

All data taken into account

Disadvantages of standard deviation:

Complicated calculation

Box and whisker diagram:

Normal distribution:

How does range, IQR, and standard deviation change if constant C is added to each datum?

No change

How does range, IQR, and standard deviation change if each datum is multiplied by constant k?

Multiplied by constant k

How does standard deviation change when deleting data far from mean?

Decrease

How does standard deviation change when deleting data close to mean?

Increases

How does standard deviation change when adding data far from mean?

Increase

How does standard deviation change when deleting data close to mean?

Decreases

How does range change when added/deleted datum affects max/min datum?

Increases/decreases

How does range change when added/deleted datum doesn’t affect max/min datum?

Unchanged

How does interquartile range change when added/deleted datum affects values of upper quartile and/or lower quartile?

Increases/decreases

How does interquartile range change when added/deleted datum doesn’t affect values of upper quartile and/or lower quartile?

No change

How does variance change if each datum is increased or decreased by constant?

No change

How does variance change if each datum is multiplied or divided by constant?

Multiplied or divided by constant squared

How does mean, mode, median change if each datum is multiplied by constant -k?

Multiplied by -k

How does range, interquartile range, and standard deviation change if each datum is multiplied by constant -k?

Multiplied by k

How does variance change if each datum is multiplied by constant -k?

Multiplied by k^2

Formula for mean in words:

Sum of data values / n

Formula for sample mean in symbols:

Symbol for population variance

σ^2

Symbol for sample variance

s^2

IQR method to determine outliers:

Lower bound = Q1 - 1.5(IQR)

Upper bound = Q3 + 1.5(IQR)

Standard deviation method to determine outliers:

Lower bound = x bar - 2(standard deviation)

Upper bound = x bar + 2(standard deviation)

Measures that are not greatly affected by outliers

Resistant / robust measures

Resistant measures of center and spread:

Median, IQR, mode