Khan Academy AP Statistics Basics (1)

 
  • 0 views

Statistics Basics (1)

1. Statistics

  • Descriptive

  • Inferential

  • 1.1 Mean, Median, Mode

    Find the mean, median and mode of the following set of numbers:

    23 29 20 32 23 21 33 25

    \[mean = sum/total\]
    • mean = 25.75
    \[median = middle number\]
    • median = (23+25)/2=24
    \[mode众数\]
    • mode = 23

  • 1.2 Range and midrange

    Find the mean, median and mode of the following set of numbers:

    65 81 73 85 94 79 67 83 82

    \[Range= Largest-Smallest\]
    • Range=94-65=29
    \[Mid-Range=(Lagrest+Smallest)/2\]
    • Mid-Range=(94+65)/2=79.5

2. Central Tendency

  • Mean
    • Arithmetic mean算数平均数
    • Geometric mean几何平均数
  • Median中位数
  • Mode众数

3. Sample and Population

  • Sample

    • \[sample \ mean = \overline x ={\sum_{i=1}^{n}{x_i}\over{n}}\]

  • Population

    • \[Population \ mean =\mu={\sum_{i=1}^{N}{x_i}\over{N}}\]

4. Dispersion 离散

  • variance方差

    • \[variance=\sigma{^2}={\sum_{i=1}^{N}{(x_i-\mu)^2}\over{N}}={\sum_{i=1}^{N}{X_i^2}\over{N}}-\mu^2\]

  • sample variance 样本方差

    • \[sample \ variance=S^2={\sum_{i=1}^{n}{(x_i-{\overline x})^2}\over{n}}\]

  • 更好的样本方差计算

    • \[unbiased\ sample\ variance=S^2={\sum_{i=1}^{n}{(x_i-{\overline x})^2}\over{n-1}}\]

  • standard deviation 标准差

    • \[standard \ deviation = \sigma = \sqrt{\sum_{i=1}^{N}{(x_i-\mu)^2}\over{N}}\]

5. Random Variable 随机变量

  • Discrete 离散随机变量–有限个数

    • \[Discrete \ Radndom \ Variable = \begin{Bmatrix} 1&if \ it \ rains \ tommorrow \\ 0&if \ not \ rains \ tommorrow \end{Bmatrix}\]

  • Continuous 连续随机变量–无限个数

    • Probability Density概率密度函数=连续函数下一定区间的面积概率

      概率密度函数

      • 概率密度函数面积必定为1

6. Expected Random Variable 随机变量期望值

  • expected random variable(sample mean) = population mean

  • calculate the sample with many times

    • 计算投掷6次骰子出现正面的概率？
    \[E(x) = n * p\qquad \\ 0 * 1.56\% + 1 * 9.37\% + 2 * 23.43\% + 3 * 31.25\% + 4 * 23.43\% + 5 * 9.37\% + 6 * 1.56\% = 3\]

    probability distribution

  • 二项式分布公式

    • \[P_{(x=k)}=C^k_n(p^k)*(1-p)^{(n-k)}\]
    • n表示试验次数
    • k表示指定事件发生次数
    • p代表指定事迹在一次试验中的发生概率

7.Poisson Distribution 泊松分布

8. Law of Large Numbers 大数定理

  • 当试验次数足够大时，样本的均值将收敛于总体均值或随机变量的期望值。

9. Normal/Gaussian Distribution 正态/高斯分布

  • 连续函数

  • 公式 \(f(x) = {1\over\sqrt{2\pi}\sigma} * e^{-{1\over2}*({x-\mu\over\sigma})^2}\)

    • μ:总体期望均值
    • σ^2:方差
    • x:随机变量

  • 累积分布函数（cumulative distribution function ）

10. z - score

11. Empirical Rule(68-95-99.7 法则)

  • The empirical rule states that for a normal distribution, nearly all of the data will fall within three standard deviations of the mean. The empirical rule can be broken down into three parts:
    • 68% of data falls within the first standard deviation from the mean.
    • 95% fall within two standard deviations.
    • 99.7% fall within three standard deviations.
  • empirical rule

12. Standard Normal Distribution标准正态分布

  • 均值μ = 0
  • 标准差σ = 1

13. Central Limit Theorem 中心极限定理

  • The Central Limit Theorem states that the sampling distribution of the sample means approaches a normal distribution as the sample size gets larger — no matter what the shape of the population distribution. This fact holds especially true for sample sizes over 30. All this is saying is that as you take more samples, especially large ones, your graph of the sample means will look more like a normal distribution.

14. Sampling Distribution of the Sample Mean 样本均值的抽样分布

  • 样本均值的抽样分布是所有的样本均值形成的分布，即μ的概率分布。样本均值的抽样分布在形状上却是对称的。

  • 随着样本量n的增大，不论原来的总体是否服从正态分布，样本均值的抽样分布都将趋于正态分布，其分布的数学期望为总体均值μ，方差(standard error)为总体方差的1/n。

  • \[\sigma^2_{\over{x}} = {\sigma^2\over{n}}\]

  • 样本量n越大，标准差越小，越趋于正态分布。

  • The standard error of a statistic is the standard deviation of the sampling distribution of that statistic. For example, the standard error of the mean is the standard deviation of the sampling distribution of the mean. Standard errors play a critical role in constructing confidence intervals and in significance testing.

15 Interval Confidence置信区间

  • A confidence interval is how much uncertainty there is with any particular statistic. Confidence intervals are often used with a margin of error. It tells you how confident you can be that the results from a poll or survey reflect what you would expect to find if it were possible to survey the entire population. Confidence intervals are intrinsically connected to confidence levels.

  • Confidence Intervals vs. Confidence Levels

    • Confidence levels are expressed as a percentage (for example, a 95% confidence level). It means that should you repeat an experiment or survey over and over again, 95 percent of the time your results will match the results you get from a population (in other words, your statistics would be sound!).
    • Confidence intervals are your results…usually numbers. For example, you survey a group of pet owners to see how many cans of dog food they purchase a year. You test your statistics at the 99 percent confidence level and get a confidence interval of (200,300). That means you think they buy between 200 and 300 cans a year. You’re super confident (99% is a very high level!) that your results are sound, statistically.
    • a level=2 tails interval

16 T-distribution vs. Z-distribution

  • T-Score vs. Z-Score

  • Degrees of Freedom

    • Degrees of freedom of an estimate is the number of independent pieces of information that went into calculating the estimate. It’s not quite the same as the number of items in the sample. In order to get the df for the estimate, you have to subtract 1 from the number of items. Let’s say you were finding the mean weight loss for a low-carb diet. You could use 4 people, giving 3 degrees of freedom (4 – 1 = 3), or you could use one hundred people with df = 99.

      • \[Degrees\ of\ Freedom(Df) = n-1\]

    • Why do we subtract 1 from the number of items?

    • Another way to look at degrees of freedom is that they are the number of values that are free to vary in a data set. What does “free to vary” mean? Here’s an example using the mean (average): Q. Pick a set of numbers that have a mean (average) of 10. A. Some sets of numbers you might pick: 9, 10, 11 or 8, 10, 12 or 5, 10, 15. Once you have chosen the first two numbers in the set, the third is fixed. In other words, you can’t choose the third item in the set. The only numbers that are free to vary are the first two. You can pick 9 + 10 or 5 + 15, but once you’ve made that decision you must choose a particular number that will give you the mean you are looking for. So degrees of freedom for a set of three numbers is TWO.

    ​

  • 样本量小(n<30)或不知道总样本σ时用T-distribution

    • T distribution

      • The T distribution (also called Student’s T Distribution) is a family of distributions that look almost identical to the normal distribution curve, only a bit shorter and fatter. The t distribution is used instead of the normal distribution when you have small samples (for more on this, see: t-score vs. z-score). The larger the sample size, the more the t distribution looks like the normal distribution. In fact, for sample sizes larger than 20 (e.g. more degrees of freedom), the distribution is almost exactly like the normal distribution.
      • T table

  • 样本量大(n>30)或知道总样本σ时用z-distribution

    • If you don’t know your population mean (μ) but you do know the standard deviation (σ), you can find a confidence interval for the population mean, with the formula:

      • \[\overline{x} ± z* {σ \over{ \sqrt{n}}}\]

16 伯努利分布（Bernoulli Distribution）

  • 伯努利分布亦称“零一分布”、“两点分布”。称随机变量X有伯努利分布, 参数为p(0<p<1),如果它分别以概率p和1-p取1和0为值。

    • \[\mu = p\]
    • \[\sigma^2 = (1-p)(0-p)^2 + p(1-p)^2=p(1-p)\]

  • p: probability of success