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STAT 515 --- STATISTICAL METHODS
Statistics: The science of using data to make decisions and draw conclusions
Two branches:
Descriptive Statistics: The collection and presentation (through graphical and numerical methods) of data Tries to look for patterns in data, summarize information
Inferential Statistics: Drawing conclusions about a large set of individuals based on data gathered on a smaller set Example:
Some Definitions
Population: The complete collection of units (individuals or objects) of interest in a study
Variable: A characteristic of an individual that we can measure or observe
Examples:
Sample: A (smaller) subset of individuals chosen from the population
In statistical inference, we use sample data (the values of the variables for the sampled individuals) to make some conclusion (e.g., an estimate or prediction) about the population.
Example:
How reliable is this generalization to the population? For inference to be useful, we need some genuine measure of its reliability.
Types of Data:
Quantitative (Numerical) Data: Measurements recorded on a natural numerical scale (can perform mathematical operations on data).
Qualitative (Categorical) Data: Measurements classified into one of several categories.
Typically, an experiment is better for establishing cause/effect between two variables, but it’s not always practical or possible.
Regardless of the type of study, we must ensure that we have a representative sample, one that has similar characteristics to the population.
The best kind of sample is a simple random sample (every subset has an equal chance of being selected)
Standard statistical methods assume the data are a random sample from the population.
Methods for Describing Data Sets
Important Principle in Statistics: Data Reduction
Example:
List of 100 numbers would be confusing, not informative Need to reduce data to a reasonable summary of information.
Two ways: Graphs and Plots Numerical Statistics
Describing Qualitative Data
Data are categorized into classes The number of observations (data values) in a class is the class frequency Class Relative Frequency = class frequency / n The CRF’s of all classes add up to 1
Example:
Histogram: Numerical data values are grouped into measurement classes, defined by equal-length intervals along the numerical scale.
Like a bar graph, a histogram is a plot with the measurement classes on the horizontal axis and the class frequencies (or relative frequencies) on the vertical axis.
For each measurement class, the height of the bar gives the frequency (or RF) of that class in the data.
Example:
Guidelines for Selecting Measurement Class Intervals: Use intervals of equal width Each data value must belong to exactly one class Commonly, between 5 and 12 classes are used
Note: Different choices of Class Intervals (in position and number) may produce different-looking histograms.
Most often, we use software to help choose intervals and create histograms.
Histograms don’t show individual measurement values (stem-and-leaf displays and dot plots do).
But for large data sets, histograms give a cleaner, simpler picture of the data.
Summation Notation
In statistics, we customarily denote our data values as x 1 , x 2 , …, x n. ( n is the total number of observations.)
The sum of a set of numbers is denoted with Σ.
That is, x 1 + x 2 + … + x n =
Or, we might sum the squared observations: x 12 + x 22 + … + x n^2 =
Example:
Example: Mark McGwire’s Home Run totals (1987- 1998). Ordered Data: 9, 9, 22, 32, 33, 39, 39, 42, 49, 52, 58, 70.
Two years with 9 home runs are outliers (unusual values) due to injury and a players’ strike.
What if we delete these years?
Which measure was more affected by the outliers?
Shapes of Distributions When the pattern of data to the left of the center value looks the same as the pattern to the right of the center, we say the data have a symmetric distribution. Picture:
If the distribution (pattern) of data is imbalanced to one side, we say the distribution is skewed.
Skewed to the Right (long right “tail”). Picture:
Skewed to the Left (long left “tail”). Picture:
Comparing the mean and the median can indicate the skewness of a data set.
Example: You want to invest in a stock for a year. Two stocks have the same average annual return over the past 30 years. But how much does the annual return vary from year to year?
Question: How much is a data set typically spread out around its mean?
Deviation from Mean: For each x -value, its deviation from the mean is:
Example (Heights of sample of plants): Data: 1, 1, 1, 4, 7, 7, 7. Deviations:
Squared Deviations:
A common measure of spread is based on the squared deviations. Sample variance: The “average” squared deviation (using n -1 as the divisor)
Definitional Formula:
s^2 =
Previous example: s^2 =
Shortcut formula: s^2 =
Another common measure of spread: Sample standard deviation = positive square root of sample variance.
Previous example: Standard deviation: s =
Note: s is measured in same units as the original data.
Why divide by n -1 instead of n? Dividing by n -1 makes the sample variance a more accurate estimate of the population variance, σ^2.
The larger the standard deviation or the variance is, the more spread/variability in the data set.
Usually use computers/calculators to calculate s^2 and s.
Rules to Interpret Standard Deviations
Think about the shape of a histogram for a data set as an indication of the shape of the distribution of that variable.
Example: Suppose IQ scores have mean 100 and standard deviation 15, and their distribution is mound- shaped.
Example: The rainfall data have a mean of 34.9 inches and a standard deviation of 13.7 inches.
What if the data may not have a mound-shaped distribution?
Chebyshev’s Rule: For any type of data, the proportion of data which are within k standard deviations of the mean is at least:
In the general case, at least what proportion of the data lie within 2 standard deviations of the mean?
What proportion would this be if the data were known to have a mound-shaped distribution?
Rainfall example revisited:
Numerical Measures of Relative Standing These tell us how a value compares relative to the rest of the population or sample. Percentiles are numbers that divide the ordered data into 100 equal parts. The p -th percentile is a number such that at most p % of the data are less than that number and at most (100 – p )% of the data are greater than that number.
Well-known Percentiles: Median is the 50th^ percentile. Lower Quartile (QL) is the 25th^ percentile: At most 25% of the data are less than QL; at most 75% of the data are greater than QL. Upper Quartile (QU) is the 75th^ percentile: At most 75% of the data are less than QU; at most 25% of the data are greater than QU.
Which score was better relative to the class’s performance?
Your friend got a 66 on the Spanish test:
z-score:
Boxplots, Outliers, and Normal Q-Q plots
Outliers are observations whose values are unusually large or small relative to the whole data set.
Causes for Outliers: (1) Mistake in recording the measurement (2) Measurement comes from some different population (3) Simply represents an unusually rare outcome
Detecting Outliers
Boxplots: A boxplot is a graph that depicts elements in the 5-number summary.
Picture:
The “box” extends from the lower quartile QL to the upper quartile QU. The length of this box is called the Interquartile Range (IQR) of the data. IQR = QU – QL The “whiskers” extend to the smallest and largest data values, except for outliers.
Defining an outlier:
If a data value is less than QL – 1.5(IQR) or greater than QU + 1.5(IQR), then it is considered an outlier and given a separate mark on the boxplot. We generally use software to create boxplots.
Interpreting boxplots
A long “box” indicates large variability in the data set. If one of the whiskers is long, it indicates skewness in that direction. A “balanced” boxplot indicates a symmetric distribution.
Outliers should be rechecked to determine their cause. Do not automatically delete outliers from the analysis --- they may indicate something important about the population.
