STSCI 2150 Prelim 1

Statistics

Measurement, variation/variability, & comparison

Goals of statistics

Estimate the values of important parameters, measure variability, compare groups, test hypotheses

Sample of convenience

Collection of individuals that happen to be available at the time; bad because study sample and population of interest should be as similar as possible

Variable

Characteristic measured on individuals drawn from a population under study

Data

Measurements of 1+ variables made on a collection of individuals

Data formatting

Rows contain observations, columns correspond to variables

Response & explanatory variable

Response variable is predicted or explained from the explanatory variable
Response~outcome; explanatory~predictor

Populations

Parameters (Greek letters); the "true" value
Include all existing organisms of interest

Samples

Estimates (Roman letters); an approximation or guess about the "truth" based on a sample group chosen methodically & randomly

Parameters vs. estimates

Population parameters are constants; estimates are random variables that change from sample to the next

Bias

Systematic discrepancy between estimates and the true population characteristic; if biased, not accurate

What makes an estimate accurate (unbiased)?

If the average of estimates obtained is centered on the true population value

Volunteer bias

Volunteers for a study are likely to be different on average from the population

How can we tell when an estimate is biased?

If experiment is repeated and estimate is consistently too high or too low

Precision

Measure of how far apart repeated estimates might be; a mathematical concept that can be derived

How do we know if estimates are precise?

Precise estimates are relatively close to each other

Effect of larger sample on precision

Larger samples yield more precise estimates

Properties of a good sample

Random selection of individuals (representative of the population of interest), independent selection of individuals, sufficiently large (large samples yield more precise estimates)

Random sample

Each member of population has an equal and independent chance of being selected

Experimental vs. observational studies

Experimental: researcher randomly assigns individuals to treatment groups; more powerful and can help determine cause-and-effect relationships
Observational: assignment of treatments is not made by researcher; can only assess associations between variables

Categorical variables

Dichotomous (binary), ordinal (ordered categories; ex. cancer stage, education level, recovery from an operation), nominal (categories have no natural ordering; ex. drug treatment, region, species type)

Numerical (quantitative) variables

Continuous (can be measured; ex. age, weight, miles traveled) or discrete (can be counted; ex. number of offspring, number of days)

Why does the type of variable matter?

Determines the way it's summarized (average vs. percentages), the type of statistical analysis, the graphical display format

Graphing categorical variables

Bar graph

Graphing numerical variables

Histogram, box plot, dot plot (usually for smaller amounts of data)

Features of a box plot

Center line in box: median
Top edge of box: 3rd quartile/75th percentile
Bottom edge of box: 1st quartile/25th percentile
Whiskers: extend to smallest and largest non-extreme observations
Extreme observations: observations beyond 1.5 IQR from box edges

Graphing two numerical variables

Scatter plot, line graph, map

Graphing two categorical variables

Mosaic plot (always preferred!), grouped bar graph

Graphing a categorical and numerical variable

Multiple (stacked) histograms, side-by-side box plots

Bad graphs

Pie charts & 3D graphs (hard to make accurate interpretations & perspective skews visual perception)

Common graphing errors

Truncation of y-axis, cumulative graphs, ignoring conventions, mislabeled or missing axes

Guidelines for good graphics

Show the data
Represent magnitudes accurately
Draw graphical elements clearly, minimizing clutter (maximize data-to-ink ratio)
Make displays easy to interpret
Clearly identify axes
All figures should have captions
Be sure that figures with color are effective in b&w

How is location (central tendency) measured?

Mean, median, mode

Sample mean

Average of observations, center of gravity; find sum of observations and divide by count

Median

Middle measurement in a set of ordered data; if even number of observations average the two middle values

Mode

Most frequent measurement

How is variability (width or spread) measured?

Range, standard deviation, variance, interquartile range

Range

Maximum value - minimum value
Poor measure of distribution width/biased estimator of true population range; small samples tend to give lower estimates of range than large samples

Sample variance

s² = (x1 - x̄ )² + (x2 - x̄ )² + ... + (xn - x̄ )² / (n-1)
(Almost) average squared difference from the mean; in original units squared

Standard deviation

s = sqrt(s²)
Sigma is the true standard deviation, s is the sample standard deviation
Related to the average distance between the mean and each observation; measures variability/spread of a distribution

Interpreting standard deviation in a normal distribution (bell curve)

2/3 (66.6%) of data falls within 1 standard deviation of the mean
95% of data falls within 2 standard deviations of the mean

Skew

A measurement of asymmetry; direction of skew refers to pointy tail of distribution

Mean vs. median in skewed data

Right-skewed data: sample mean > sample median
Left-skewed data: sample mean < sample median

Estimating standard deviation from a histogram

Eyeball 2.5% of sample size from top (U) and bottom (L)
s ~= (U - L) / 4

Does repeating an experiment result in the same results?

No, because of random variation (especially with small samples)

What is the key concept behind a sampling distribution, confidence intervals, and standard errors?

Variability and uncertainty of samples (and sample means)

Effect of increasing sample size on spread and variation

Larger sample size reduces the spread/variation of the sampling distribution of an estimate

Sampling distribution

Probability distribution of all values for an estimate we might have obtained when the population was sampled; illustrates how much the sample mean could vary and what values are typical
Only known in theory or via simulations

Standard error

Quantifies the innate variability of an estimator (uncertainty); the standard deviation of the estimator's sampling distribution

Standard error of the mean

SE = s/sqrt(n)
Sigma notation rarely used because the true standard deviation is rarely known, in most cases we only have a sample

Is the sample mean a good estimator of the population mean?

Yes

Do larger samples fix bias?

No

Confidence interval

Quantifies uncertainty about a parameter; an interval estimate of plausible values (not point) for the true population mean

How is a 95% confidence interval worded?

We are 95% confident that the true population value lies within the interval...
If we drew repeated samples, we expect that about 95% of the confidence intervals would contain the true population value
(NOT probability)

2 standard error rule of thumb

Assuming a normally distributed population and/or sufficiently large sample size, the interval of 2 standard errors above and below the mean roughly estimate the 95% confidence interval for the mean

Confidence intervals of different samples...

Different samples yield different intervals
Intervals vary in width
~95% of the time contains true parameter value, ~5% of time doesn't (but in reality we never know)

Effect of increasing sample size on confidence intervals

Larger sample size yields narrower confidence intervals

99% confidence intervals vs. 95% confidence intervals

99% intervals are wider than 95% intervals

Distribution of the data

Set of all values in sample