AP Stat Ch 1 Flashcards
Available data
The data that were produced in the past for some other purpose but that may help answer a present question
Statistics
The science of collecting, analyzing, and drawing conclusions from data
Observational study
In an observational study, we observe individuals and measure variables of interest but do not attempt to influence the responses
Experiment
In an experiment, we deliberately do something to individuals in order to observe their responses
Individuals
Individuals are the objects described by a set of data. Individuals may be people, but they may also be animals or things. Do not get individuals confused with the population
Population
The population of interest is the entire collection of individuals or objects about which information is desired
Variable
Any characteristic of an individual whose value may change from one individual to another.
Ex. Hair color, height, brand of car, gpa
Categorical variable
An individual into one of several groups or catergories.
Ex. Hair color, brand of car
USUALLY WORDS AS OPTIONS
Quantitative
Numerical data. Takes numerical values for which arithmetic operations such as adding and averaging make sense.
Categorical vs quantitative variables
Categorical is w words whereas quantitative is with numbers–can do operations to them
Census
When you study an entire population, it is called a census
Sample
A sample is a subset of the Population, selected for study in some prescribed manner
Descriptive statistics
The branch of statistics that includes methods for organizing and summarizing data
Inferential statistics
The branch of statistics that involves generalizing about a population based on information from a sample of that population.
Statistical inference
The process of drawing these generalizations about inferential statistics
Distribution of a variable
Tells us what values the variable takes and how often it takes these values
Discrete data
Quantitative data is discrete if the possible values are isolated points on the number line.
Shoe size, number of birthdays. Count them. Whole numbers.
Continuous data
Numerical data is continuous if the possible values form an entire interval on the number line
Foot length, age
Discrete vs continuous variables
Measure continuous, count discrete
Types of variables
First decide if Categorical or quantitative.
If catergorical, then it is words– hair color, fav color, fav president
If quantitative then it is numbers – age, number siblings
If quantitative, then discrete or continuous
Discrete if u can count it, continuous if u measure it.
Discrete is number of pages, continuous is length of an inseam
Are the following quantitative (continuous or discrete) or caterogircal: Length of pen Color of pants Subject of book Type of pen Number of pockets Number of pages Number of pens in a box Length of an inseam Area of a page
Length of pen– quantitative, continuous
Color of pants– caterogircal
Subject of book– cateofgircal
Type of pen– cateorgircal
Number of pockets– quantitative, discrete
Number of pages– quantitative, discrete
Number of pens in a box – quantitative, discrete
Length of inseam– quantitative, continuous
Area of a page– quantitative, continuous
Frequency table
For caterogircal data, make a frequency table – displays the possible catergories and either the count or the present of individuals who fall in each category
Frequency
Count– # of items in that group
Relative frequency
Percent of your thing. If you have 2 and there are 11 total, relative frequency = 2/11
Ways to display caterogircal data
Bar graphs and relative frequency bar graphs
Pie charts and segments bad charts
Two way table
Bar graphs and relative frequency bar graphs
Label variables and scales
The bars should be the same width and not touching each other
The order of the categories doesn’t matter
Relative frequency bar charts make it easier to compare multiple distributions, especially when the sample sizes are different
Pie charts and segmented bar charts
Label variables and categories
Pie charts are easier to construct with a computer spreadsheet program or stat software
Pie charts help us visually see what part of the whole each group forms
Segmented bar charts are basically rectangular pie charts, each bar is a whole, divide each bar proportionally into segments corresponding to the percentage in each group
Segmented bar charts make it easier to compare distributions
AP exam common error with charts
BE SURE TO LABEL GRAPHS!!!
Suppose I wanted to compare AP stat scores for tenth, eleventh, and twelfth graders. Which type of graph would be the best?
Segmented bar chart
Three bars, one with tenth, one with eleventh, one with twelfth
Two way table
A table with two categorical variables
Marginal distribution
Distributions of categorical data that appear at the right and bottom margins of a two way table. They help us to look at the distribution of each variable separately
Conditional distributions
Caterogiral distrivutions inside a two way table that deals w a specific number inside the table
How many total conditional distributions are there?
Rows + columns
Simpson’s paradox
An association between two variables that holds for each individual value of a third variable can be changed or even reversed when the data for all values of the third variable are combined. This reversal is called Simpson’s paradox. Therefore You must be careful when data from several groups are combined to form a single group!
Data that suggests one conclusion when aggregated and a different conclusion when presented in subcategories
Lurking variables
With Simpson’s paradox
Sometimes the relationship between two variables is influenced by other variables that we did not measure or even think about! Because the variables are lurking in the background, we call them lurking variables. They are not among the explanatory or response variables in a study, but they may influence the interpretation of the relationship among these variables.
Conclusions from Simpson’s paradox
It is caused by a combination of a lurking variable and data from unequal sized groups being combined into a single data set. The unequal group sizes, in the prescense of a lurking variable, can weight the results incorrectly. This can lead to seriously flawed conclusions. The obvious way to prevent it is to not combine data sets of different sizes from diverse sources!
A great deal of care has to be taken when combining small data sets into a larger one.
Sometimes Conclusions from large data sets are the opposite of conclusions from smaller ones. Conclusions from large set are usually wrong!
Dotplot
A simple way to display quantitative data when the set is reasonably small
How to construct a dotplot
Label your axis (horizontal line) with the variable and title your graph
Scale the axis based on the values of the variable
Mark a dot above the number on the horizontal axis corresponding to each data value. Stack multiple dots vertically
Stem and leaf plot
Another way to display a relatively small numerical data set. Often the values of the variable are too spread out to make a dotplot, so this is a better option. Stem is the first part of the number and leaf is last digit
How to construct a stem plot
Separate each observation into a set consisting of all but the rightmost digit and a leaf, the final digit.
Write the stems vertically in increasing order from top to bottom, and draw a vertical line to the right of the stems.
Write each leaf to the right of its stem.
Numbers to the left of the line are the stems and to the right are the leaves.
MUST INCLUDE A KEY W UNITS
LEAVES MUST BE IN SINGLE DIGITS, NO COMMAS
it is best if leaves are in numerical order
Back to back stemplots
Useful for comparing distributions
Example is comparing female and male weights
Have stem in the middle and leaves on both sides with male above one side and female above the other
Split stemplot
When a data set is very compact, it is often useful to split stems to stretch the display to investigate the shape. Whenever you split stems, be sure that each stem is assigned an equal number of possible leaf digits.
When given data all between 96 and 99, make stems 96,96,97,97,98,98,99,99 and have the top be 0-4 for leaves and bottom be 5-9
What to do when data is spread out for stem and leaf plot
Truncate or round the data to shrink the display
Change 10.53 to 11
Describing a distribution
SHAPE, CENTER, AND SPREAD
Shape
Symmetric, skewed right, or skewed left
Unimodal if one peak, bimodal if two peaks
Uniform if a plateau, get same values
Outliers
Data values that fall outside the overall pattern of the rest of the distribution.
<Q1 - 1.5IQR or >Q3 + 1.5IQR
Clusters
Isolated groups of points of points
Gaps
Large spaces between points
Symmetric
If the right and left sides of the historgram are approximately mirror images of each other
Skewed
The thinner ends of a distribution are called the tails. If one tail stretches out further than the other, the historgram is said to be skewed to the side of the longer tail
Histogram
Used to display larger data sets for quantitative data
Discrete histogram vs continuous histogram
In discrete historgrams, make the bars over the center of the number on the X-axis.
In continuous histograms, make classes where the bars fall between. For example, make groups of 5 and have on the left edge 40, right edge 45 and then 50 and then 55.
How to make histograms
Label axis and scales
Bars should touch
Y axis is frequency or relative frequency
X axis is variable
Relative frequency histograms
Same as regular histogram, but have relative frequency (percent of total) rather than frequency (number of observations) on the vertical axis. Relative frequency histograms are more useful because you can compare two distributions easier
Histograms vs bar graphs
Histograms uses QUANTITATIVE variables while bar graphs use CATEGORICAL data. Histograms don’t have spaces between bars, bar graphs have spaces
Continuous histograms
Make classes of the same length that never overlap Divide the range of the data into classes of equal width. Count the number of observations in each class Five classes is a good minimum. Too few will give a skyscraper graph and too many will give a pancake graph. Label and scale your axes If an observation falls on a boundary, put the value into the upper class.
Ogive
Culumative relative frequency graph
Relative culm frequency is percentile
Measuring the center of a data set
Look at the mean and the median
Population mean
Greek letter mu (u with long stem)
The arithmetic average of all values in the entire population
Sample mean
X with a bar above it.
Since we rarely study the entire population, estimate population mean with the sample mean
= sum of all values / number of values
Median
The middle score
To find which value is the middle score, put all the data in order
Mode
Most frequency observation. Not a useful measure of center.
Resistant measure
Measure not affected by outliers
Are median and mean resistant?
Median is resistant–not affected by outliers so it is better for a skewed data set
Mean is not resistant–affected by outliers, as outliers affect arithmetic average.
When to use mean and when to use median
Use median with all data
Mean with symmetrical data since mean is not resistant and median is
Skewed right vs skewed left
Skewed left is when the tail is to the left.
Median> mean
Lower values that push the graph to the left.
Bell curve on right. Tail on left.
Skewed right is when the tail is to the right
Mean>median
Bell curve on the left.
Mean vs median in skewed data
Skewed left:
Median> mean
Skewed right:
Mean> median
Symmetric:
Mean roughly equal to median
Range
Full spread of data by simply finding the difference between the largest and the smallest observation. ONE NUMBER
MAX-MIN
BUT it is not resistant. Outliers heavily influence the range
How to measure spread
Range for roughly symmetric data without outliers.
IQR when skewed or have outliers
Inter-Quartile Range
A resistant measure of spread. It is the distance between the first and third quartiles. The range of the middle half of the data.
IQR=Q3-Q1
Quartiles
Q1 is first quartile–the point that divides the lowest 25% of the data from the upper 75%
Q2 is the median
Q3 is the third quartile–the point that divides the lowest 75% of the data from the upper 25%
How to find quartiles
Get data in order. Find median. Median is Q2
Half the data above the median is Q3 and half the data below the median is Q1
In that data above the median, take that median. That value is Q3. Do the same for the data below the median and get Q1
Shape center spread summary
Shape:
Skewed (direction) or symmetric
Unimodal or bimodal
Center:
Mean or median
Spread:
IQR, range, standard deviation
Five number summary
(MIN,Q1,MED,Q3,MAX)
Box plot
A graph of the five number summary. Easy to make and clearly shows center and spread of the distribution. Skewed toward the side with the longer box.
Useful to compare multiple distributions – side by side boxplots and are usually drawn vertically
Drawing a box plot
Central box spans the quartiles Q1 and Q3
A line in the box marks the Median, M
Lines extend from the box out to the smallest and largest observations.
Width of the box = IQR
label axes and scale
Modified boxplot
Specifically identifies outliers, in addition to median and quartiles. Regular boxplot connects outliers.
Variance
Averaged squared deviation of the observations from the mean
S squared
Deviation
The deviation of an observation is its distance from the mean (x-x bar). The mean is the point that makes the sum of the deviations=0. We square the deviations to make negatives positives.
Population standard deviation
Greek letter looking like the letter “o’
Standard deviation of all the values in the entire population. Typical deviation from the mean or the average distance form the average
= SQRT (sum of (x-mu)^2 / n)
Population variance
Square of the population standard deviation
Greek letter looking like “o” squared
Sample standard deviation
Represented by s
Since we rarely study entire populations, use this.
Your distance from the center or your average distance from the average.
Approximates the average, or typical deviation
= SQRT ( sum of (x-x bar)^2 / (n-1))
Sample variance
Square of the sample standard deviation.
S squared
Why do we divide by n-1 when calculating the sample standard deviation
Some error between x bar and mu, so this helps to accounts for this.
When to use sample and when use population
Use sample unless told otherwise
When to use sample standard deviation
When talking about mean, as this measures spread about the mean.
When does s=0
When there is no spread. All observations have same value. Otherwise, s>0
As observations are more spread out about their mean, s is larger.
Is s resistant?
No because like the mean
Strong skewness or outliers can make S very large
5 number summary
Usually better than the mean and standard deviation for describing a skewed distribution or a distribution with strong outliers
Use the mean and s when with reasonably symmetric disturbition free of outliers.
Transformations to lists
Shape never changes.
Center always changes – when multiplying each observation by b, multiply both mean and median by b. Adding same number a adds a to mean and median
Spread stays the same when adding same amount to each but increases if multiply each data point by something. When milt by b, spread is multiplied by b.
