Statistics find information about group {population, statistics} {statistical population}. Population has number of people or objects, which have measurable properties {statistic, data} {descriptive statistic} {datum}. Experimenters can check population sample or collect information from all population members {census, population}.
If many random same-size samples come from a population with normally distributed measurements, sample sums and means are both normal distributions {central limit theorem, population}.
Samples have numbers {degrees of freedom, statistics}| of independent values, free to change. Degrees of freedom equal value number minus one, because last value is total minus other values and so is dependent. If situation has factors, factor degrees of freedom are factor number minus one, because last factor depends on the others.
Taking {sampling}| one value {sample, population} from a population can be random {random sample}. Sample sets {aggregate, population} have number of samples {sample size}.
attributes
Samples have properties {parameter, population} {sampling statistic}, such as weight. Samples have property values {attribute data} {measurement data}, such as high weight. Sampling can infer {statistical inference} population mean, variance, mode, and median.
types
Random samples can have same subgroup proportions as population {stratified random sample}, such as same age distribution.
Sampling can return sampled member to population {sampling with replacement} or not {sampling without replacement}.
subsets
Sample subsets {sample class} share an attribute, such as high weight.
Data can have linear changes over time {trend}| {trend line}.
Numbers can have coefficients {weight, multiplier} based on frequency or importance {weighting}|. Averages {weighted average} can use weights.
Means can have an estimate {estimate} {estimation}|.
Intervals {confidence interval}| around estimated means have a confidence level that population mean is in the interval. Unbiased estimate can have 95% confidence if low estimate equals x - (1.96 * s) / N^0.5 and high estimate equals x + (1.96 * s) / N^0.5, where x is mean, s is standard deviation, and N is sample size.
On average, situations have sets of expected outcomes {expectation, statistics} {expected value}|. Averaging outcome values can find expected value: sum from i = 1 to i = N of (w(i) * p(i)) / N, where N is number of values, w(i) is outcome worth, and p(i) is outcome probability.
Average is not the best estimate {Stein's paradox} {Stein paradox}. Best estimate is average plus factor times difference between average and grand average: e = u * f * (u - U). Factor depends on standard deviation, as shown by Bayes.
Estimates have population, which sets correct confidence interval {unbiased interval}. Sample means are unbiased population-mean estimates. Sample variances are unbiased population-variance estimates.
Percent of numbers that are less than a number plus half of percent of numbers equal to the number is a statistic {percentile rank} {percentile}|. For example, if n is at 50th percentile {second quartile, percentile} {fifth decile}, half of all values are less than or equal to n.
Dispersion {inter-quartile range} can be between lower quartile 25% and upper quartile 75%. Median splits inter-quartile range.
Value has cumulative probability {quantile} {inverse density function} {percent point function} for that value and all lower values. Quantile function is integral of probability function from minimum to value.
For distributions, one quarter {quartile} goes from 0th to 25th percentile {first quartile}. One quarter goes from 25th to 50th percentile {second quartile, distribution}. One quarter goes from 50th to 75th percentile {third quartile}.
Deviation {quartile deviation} (Q) can be half the difference between first and third quartiles.
Population measurement, such as weight, has calculated numbers {statistic, population}, such as median, mode, mean, and range.
Standard deviation divided by mean {coefficient of variation} {coefficient of variability} {variation coefficient} can measure population variation.
Sum from i = 1 to i = N of n(i) /N, where N equals number of numbers, and n(i) equals number, is a statistic {mean, population}| {average}. For example, the numbers 1, 2, 2, 3, 4, 5, and 6 have mean equal to (1 + 2 + 2 + 3 + 4 + 5 + 6) / 7. Average is number-group balance point, because sum of differences between numbers and mean equals zero.
If numbers are in sequence, middle number of odd number of numbers, or average of two middle numbers of even number of numbers, is a statistic {median, population}|. For example, the numbers 1, 2, 2, 3, 4, 5, and 6 have median equal 3.
Number with greatest frequency is a statistic {mode, population}|. For example, the numbers 1, 2, 2, 4, 5, and 6 have mode equal 2.
Difference between lowest and highest number is a statistic {range, number}. For example, the numbers 1, 2, 2, 3, 4, 5, and 6 have range equal 5.
Mean can divide into quotient error {relative error, statistics}.
Number sets have variance spread {dispersion, statistics}|. Dispersion is torques of numbers around balance point: sum from i = 1 to i = N of (n(i) - x)^2 / N, or sum from i = 1 to i = N of (n(i))^2 / N - x^2, where n(i) are numbers, and x equals mean.
Fourth moment {kurtosis, distribution} measures distribution fatness or slimness.
Square root of mean of squares of differences between numbers and mean {root mean square} (RMS) can equal standard deviation.
Third moment {skewness, distribution} measures distribution asymmetry, whether it is more to right or left of mean. Skew distribution is not symmetric. To find skewness, calculate median and compare to mean.
Sample-mean distribution standard deviation {standard error of the mean}| is smaller than population standard deviation: s / N^0.5, where s is population standard deviation, and N is sample size.
Variance has a square root {standard deviation}|.
Torques of numbers around balance point measure dispersion {variance, distribution}|: sum from i = 1 to i = N of (n(i) - x)^2 / N, or sum from i = 1 to i = N of (n(i))^2 / N - x^2, where n(i) are numbers, and x equals mean.
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Date Modified: 2022.0225