Statistical Measures: Location, Variance, and Probability
Measures of Location
Minimum: Smallest number in the data set.
Maximum: Largest number in the data set.
Median: The middle number, or the average of the two middle numbers if the data set has an even number of values.
Mean: The average of all numbers in the data set.
Mode: The most frequent number. A data set can have multiple modes.
Quartiles
First Quartile (Q1): The median of the lower half of the data.
Third Quartile (Q3): The median of the upper half of the data.
Measures of Variance
Range: Maximum – Minimum.
Interquartile Range (IQR): Q3 – Q1. This represents the range of the middle 50% of the data.
Standard Deviation: A measure of the spread of the data around the mean.
Calculating Variance and Standard Deviation
- Calculate the variance for each number: (Number in Data Set – Mean)2.
- Sum all the calculated variances.
- Divide by the number of data points (n) or by (n-1) if representing a sample of a population.
- Calculate the Standard Deviation: Take the square root of the variance.
Confidence Interval Calculation
- Calculate the Mean and Standard Deviation of a population.
- Select a Confidence Interval (e.g., 90%, 95%, 99%).
- Use the equation: Za/2 * σ/√(n)
Where Za/2 is the confidence coefficient, ‘a’ is the confidence level, σ is the Standard Deviation, and ‘n’ is the sample size.
Convert the confidence level percentage to a decimal, and use a z-table to find the corresponding value (e.g., 1.96 for 95%).
Finding the Standard Error and Margin of Error
- Standard Error: Divide the Standard Deviation by the square root of the sample size.
- Margin of Error: Multiply the critical value (Za/2) by the Standard Error.
- Confidence Interval: Mean ± Margin of Error.
Bernoulli Calculation: Use the binompdf function (found under DISTR in many calculators). Parameters: binompdf (number of trials, probability of occurrence, number of specific events).
Additive Property: Combining probabilities of A and B.
P(A or B) = P(A) + P(B) – P(A and B)
Five-Number Summary: Minimum, Lower Quartile (Q1), Median, Upper Quartile (Q3), Maximum.
Odds: Number of favorable outcomes divided by the number of unfavorable outcomes.
Additional Statistical Concepts
Quantiles: Decimal form of percentiles.
Measures of Location: Include Minimum, Maximum, Midrange, Median, Mean, Quartiles, and the Five-Number Summary.
Measures of Variation: Include Range, IQR, and Standard Deviation.
Understanding Standard Deviation
Standard Deviation measures the amount of variance or dispersion from the average.
- Low Standard Deviation: Data points tend to be close to the mean.
- High Standard Deviation: Data points are spread out over a larger range.
- 68-95-99.7 Rule: Approximately 68% of data falls within 1 standard deviation, 95% within 2, and 99.7% within 3 in a normal distribution.
Variance: Measures how spread out a data set is. A variance of zero indicates an identical data set.
Normal Distribution: Data tends to be focused around a central value with no bias, forming a “bell-shaped” curve. In a perfect normal distribution, Mean = Median = Mode.
Parameters of Normal Distribution: Mean and Standard Deviation. The standard deviation determines the width of the curve, and the mean determines its height.
Confidence Interval: A range around a measurement that conveys its precision.
Standard Error: Measures the accuracy with which a sample represents a population. Lower standard error means less spread in the sampling distribution.
Bernoulli Trial: A random experiment with exactly two possible outcomes: success and failure.
Mutually Exclusive Events: Events that cannot happen at the same time.
Exhaustive Events: A set of events where at least one event must occur; they cover all possibilities.
Conditional Probability: The probability that an event will occur given that another event has already occurred.