STA 291 LEC 1&2

Methods for Collecting , Describing, Analyzing and Drawing Conclusions from Data

• total set of all subjects of interest
• the entire group of people, animals or things about which we want information

any individual member of the population

• subset of the population from which the study actually collects information
• used to draw conclusions about the whole population

• a characteristic of a unit that can vary among subjects in the population/sample
• Examples: gender, nationality, age, income, hair color, height, disease status, company rating, grade in STA 291, state of residence

listing of all the units in the population

• numerical characteristic of the population
• calculated using the whole population

• numerical characteristic of the sample
• calculated using the sample

• Accuracy: May not be able to list them all - may not be able to come up with a frame
• Time: Speed of Response
• Expense: Cost
• Infinite Population
• Destructive Sampling or Testing

Summarizing the information in a collection of data

Using information from a sample to make conclusions/predictions about the population

Consists of observations on a single attribute

Consists of observations on several attributes

Two attributes collected per observation

• have a scale of unordered categories
• Examples: gender, nationality, hair color (it doesn't make sense to say that green eyes are greater/higher/better than brown

• have a scale of ordered categories; often treated in a quantitative manner
• Examples: disease status, company rating, grade in STA 291

• categorical (not numerical)
• Nominal and Ordinal

measured numerically, that is, for each subject a number is observed

the scale for quantitative variables

• has a finite number of possible values
• all qualitative (categorical) variables are ~
• only some quantitative (numeric) variables are ~

• can take all the values in a continuum of real values
• Examples: time, distance, volume, speed, (usually physical measures)

• Each possible sample has the same probability of being selected
• The sample size is usually denoted by "n"

the people just happened to be there

• this sample will poorly represent the population
• will cause misleading conclusions
• BIAS
• people are much more likely to speak up if they feel strongly about the issue
• Examples: Mall interview, Street corner interview

even if it is smaller it is much more trustworthy than volunteer because it has less bias

• observes individuals and measure variable of interest but does not attempt to influence the responses
• passive data collection
• it's purpose is to describe/compare groups or situations

• deliberately imposes some treatment on individuals in order to observe their responses
• active data production
• it's purpose is to study whether the treatment causes a change in the response

• divide the population into separate, non-overlapping groups ("strata")
• select a simple random sample independently (and usually proportionally) from each group

• the population can be divided into a set of non-overlapping subgroups (the clusters)
• the clusters are then selected at random, and all individuals in the selected clusters are included in the sample

• an initial name is selected at random
• every Kth name is selected after that
• K is computed by dividing membership list length by the desired sample size
• not a simple random sample, but often almost as good as one
• useful when the population consists as a list

• Selection Bias - selection of the sample systematically excludes some part of the population of interest
• Measurement/Response Bias - method of observation tends to produce values that systematically differ from the true value
• Nonresponse Bias - occurs when responses are not actually obtained from all individuals selected for inclusion in the sample

• the error that occurs when a statistic based on a sample estimates or predicts the value of a population parameter
• in random samples, the sampling error can usually be quantified

• any error that could also happen in a census
• Examples: bias due to question wording, question order, non-reponse, wrong answers (especially to delicate questions)

• a listing of intervals of possible values for a variable AND a tabulation of the # of observations in each interval
• -use intervals of same length (if possible)
• -intervals must be mutually exclusive (any observation must fall into one and only one interval
• - RULE of thumb: if you have n observations, the # of intervals should be about √n

• frequency = # in interval
• relative frequency = frequency/total #
• percentage = relative frequency x 100%

# of observations that fall in the class and in smaller classes

• use numbers from the frequency distribution to create a graph
• draw a bar over each interval, the height of the bar represents the relative frequency for that interval
• bars should be touching; i.e., equally extend the width of the bar at the upper and lower limits so that the bars are touching

• the bars are usually separated to emphasize that the variable is categorical rather than quantitative
• for nominal variables (no natural ordering), order the bars by frequency, except possibly for a category "other" that is always last
• for ordinal data classes are presented in the natural order, (A, B, C...)

• write the observations ordered from smallest to largest
• each observation is represented by a stem (leading digit(s)) and a leaf (final digit)
• looks like a histogram sideways - gives individual values
• contains more information than a histogram, because every single measurement can be recovered

• center, spread (numbers later)
• symmetric distributions - bell-shaped or U-shaped
• not symmetric distributions - left-skewed or right-skewed

• number of subjects observed at all the combinations of possible outcomes for the 2 variables
• ~ are identified by their number of rows and columns - a table with 2 rows and 3 columns is called a 2x3 table

• present large data sets concisely and coherently
• can replace a thousand words and still be clearly understood and comprehended
• encourage the viewer to compare two or more variables
• do not replace substance by form
• do not distort what the data reveal
• have a high "data-to-ink" ratio

• don't have a scale on the axis
• have a misleading caption
• distort by stretching/shrinking the vertical or horizontal axis
• use histograms or bar charts with bars of unequal width
• are more confusing than helpful

sample-to-sample differences

some portion of the population is not sampled at all or has a smaller representation in the sample than it has in the population

• Mean: arithmetic average
• Median: midpoint of the observations when they are arranged in increasing order
• Mode: most frequent value

stragglers that stand off away from the body of the distribution

• sample mean = x-bar
• population mean = mu
• - sometimes the mean is calculated for ordinal variables, but this doesn't always make sense (GPA = 3.8)
• it is highly influenced by outliers

• falls in the middle of the ordered sample, it n is even, average the 2 middle values
• for skewed distributions, it is more appropriate measure of central tendency than the mean (better describes a "typical value")
• it may be too insensitive to changes in the data

• compromise between the median and the mean
• 1. order the data from smallest to largest
• 2. delete a selected number of values from each end of the ordered list
• 3. find the mean of the remaining values

the percentage of values that have been deleted form each end of the ordered list when calculating the mean.

• the most frequently occurring value
• on a histogram it would be the highest bar
• it may not be unique

• variance, standard deviation
• interquartile range
• range

• 50th percentile = median
• 25th = lower quartile = Q₁
• 75th = upper quartile = Q₃

• the difference between upper and lower quartile
• IQR = Q₃ - Q₁
• range of values that contains the middle 50% of the data
• IQR increases as variability increases

reports its median, quartiles, and extremes (maximum and minimum)

• basically a graphical version of the five-number summary (unless there are outliers)
• it consists of a box that contains the central 50% of the distribution (from lower quartile to upper quartile)
• a line within the box that marks the median
• lines at 1.5 IQR's from lower/upper quartiles
• whiskers that extend to the max and min, unless there are outliers

the difference between the extremes (max/min)

• the average of the squared deviations (s²)
• ∑(x_i - ẍ)²
• n - 1
• of the population
• ∑(xi - μ)²
• N

• of the population √σ²
• of the sample √s²

• about 68% of the data are within one standard deviation from the mean
• about 95% of the data are within two standard deviation from the mean
• about 99.7% of the data are within three standard deviation from the mean