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Statistics & Probability
Averages, spread, counting and probability.
经验教训
mean of 4, 8, 15, 16, 23, 42
Introductory
Describing data with graphs
Stemplots, histograms, box plots, and the quartiles behind them.
median of 3, 1, 4, 1, 5, 9, 2, 6
Introductory
Mean, median and mode
Centre and spread: mean, median, mode, range, variance, standard deviation.
stats 2, 4, 4, 4, 5, 5, 7, 9
Core
Probability
Events, complements, independence, the addition and multiplication rules.
1/6 * 1/6
Core
Counting: permutations and combinations
Factorials, ordered and unordered selections.
5 choose 2
Core
Discrete random variables
Probability distributions, expected value, and the binomial, geometric and Poisson families.
10 choose 3 * (1/2)^3 * (1/2)^7
Core
Continuous random variables
Density functions: probability is area, and the uniform and exponential distributions.
integrate 1/10 dx from 2 to 5
Core
The normal distribution
The bell curve, z-scores, and the 68-95-99.7 rule.
integrate e^(-x^2/2)/sqrt(2*pi) dx from -1 to 1
Core
The central limit theorem
Why averages are normal: the sampling distribution of the mean.
10/sqrt(25)
Core
Confidence intervals
Estimate ± margin of error, and what “95% confident” actually claims.
50 + 1.96*10/sqrt(100)
Core
Hypothesis testing
Null and alternative hypotheses, test statistics, p-values, and the two kinds of error.
(52 - 50)/(10/sqrt(25))
Advanced
Comparing two samples
Differences of means and proportions, and paired samples.
sqrt(4^2/20 + 3^2/25)
Advanced
Chi-square tests
Goodness of fit, independence and homogeneity: comparing observed counts with expected ones.
(20-25)^2/25 + (30-25)^2/25
Core
Linear regression and correlation
The least-squares line, the correlation coefficient, and prediction.
line through (1,2) and (3,6)
Advanced
ANOVA and the F distribution
Comparing several means at once by comparing variances.
variance of 2, 4, 4, 4, 5, 5, 7, 9
Summarise data honestly: mean, median, mode, variance and standard deviation, each computed in the open. Counting principles and combinations feed straight into probability.
更多经验教训
Probability
Statistics
Describing data
Inference
其中一些经验教训从开放文本中改编: OpenStax Contemporary Mathematics, Wikipedia. 每页课目都列出其来源和执照。
此处所用符号
输入完整定义、图片和其中每个字母的含义的符号。
人们对人们提出的问题
Mean or median, which should I use?
Median when the data have outliers or a long tail (incomes, house prices); mean when the data are roughly symmetric and you want every value to count. Report both if they disagree. The gap is itself information.
What does a p-value actually say?
The probability of seeing data at least this extreme if the null hypothesis were true. It is not the probability that the null hypothesis is true.
Why divide by n − 1 for the sample variance?
The sample mean sits closer to the sample than the true mean does, so squared deviations from it are slightly too small on average; dividing by n − 1 instead of n corrects the bias.
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