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Graphing Data 4
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IntroductionLecture1.1
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Basic ChartsLecture1.2
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Basic Charts Part 2Lecture1.3
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RandomnessLecture1.4
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Mean and Standard Deviation 5
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IntroductionLecture2.1
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Random WalkLecture2.2
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Random Walk Part 2Lecture2.3
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Mean and Standard DeviationLecture2.4
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Mean and Standard Deviation Part 2Lecture2.5
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Distributions 6
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IntroductionLecture3.1
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PDF and CDFLecture3.2
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PDF and CDF Part 2Lecture3.3
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Z ScoresLecture3.4
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PlottingLecture3.5
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SolutionLecture3.6
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Correlation and Linear Regression 7
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IntroductionLecture4.1
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BasicsLecture4.2
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CorrelationLecture4.3
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Z-ScoreLecture4.4
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Z-Score Part 2Lecture4.5
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Linear RegressionLecture4.6
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Linear Regression Part 2Lecture4.7
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Probability 3
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IntroductionLecture5.1
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Probability BasicsLecture5.2
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ProbabilityLecture5.3
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Counting Principles 3
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IntroductionLecture6.1
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Factorials and PermutationsLecture6.2
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Binomial TheoremLecture6.3
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Binomial Distribution 3
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IntroductionLecture7.1
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Binomial DistributionLecture7.2
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Binomial Distribution Part 2Lecture7.3
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Confidence Interval 7
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IntroductionLecture8.1
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AlphaLecture8.2
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Confidence IntervalsLecture8.3
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Confidence Intervals Part 2Lecture8.4
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Confidence Intervals Part 3Lecture8.5
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Confidence Intervals Part 4Lecture8.6
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Confidence Intervals Part 5Lecture8.7
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Proportion Confidence Interval 3
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IntroductionLecture9.1
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EquationsLecture9.2
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SolutionLecture9.3
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Hypothesis Testing 5
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IntroductionLecture10.1
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The Null HypothesisLecture10.2
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The Null Hypothesis Part 2Lecture10.3
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One Tail TestLecture10.4
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One Tail Test Part 2Lecture10.5
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Comparing Two Means 5
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IntroductionLecture11.1
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Known Standard DeviationsLecture11.2
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Unknown Standard DeviationsLecture11.3
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Same Unknown Standard DeviationsLecture11.4
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SolutionLecture11.5
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Chi-squared Test 3
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IntroductionLecture12.1
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BasicsLecture12.2
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Expected ValuesLecture12.3
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Mean and Standard Deviation
Let’s formally define the mean equation first.
Equation
$$\overline{x} = \frac{\sum_{i=1}^{n}x_i}{n}$$
$$\overline{x} = \text{Mean of Distrbution}$$
$$ x_i = \text{The ith value of x}$$
$$ n = \text{Number of Data Points} $$
$$\overline{x} = \text{Mean of Distrbution}$$
$$ x_i = \text{The ith value of x}$$
$$ n = \text{Number of Data Points} $$
And now, let’s define standard deviation. There are two different versions, the population standard deviation and the sample standard deviation. The population is supposed to be used when we know the entire population, where as the sample is used when we only have a sample of the population. These are both the square root of their respective variances…the variances equal:
Equation
$$\sigma^2 = \frac{\sum_{i=1}^{n}(x_i-\overline{x})^2}{n}$$
$$\sigma^2 = \text{Population Variance}$$
$$\overline{x} = \text{Mean of Distrbution}$$
$$ x_i = \text{The ith value of x}$$
$$ n = \text{Number of Data Points} $$
$$\sigma^2 = \text{Population Variance}$$
$$\overline{x} = \text{Mean of Distrbution}$$
$$ x_i = \text{The ith value of x}$$
$$ n = \text{Number of Data Points} $$
Equation
$$s^2 = \frac{\sum_{i=1}^{n}(x_i-\overline{x})^2}{n-1}$$
$$s^2 = \text{Sample Variance}$$
$$\overline{x} = \text{Mean of Distrbution}$$
$$ x_i = \text{The ith value of x}$$
$$ n = \text{Number of Data Points} $$
$$s^2 = \text{Sample Variance}$$
$$\overline{x} = \text{Mean of Distrbution}$$
$$ x_i = \text{The ith value of x}$$
$$ n = \text{Number of Data Points} $$
The only difference is the n is n-1. For standard deviation, we just need the square root.
Equation
$$\sigma = \sqrt{\frac{\sum_{i=1}^{n}(x_i-\overline{x})^2}{n}}$$
$$\sigma = \text{Population Standard Deviation}$$
$$\overline{x} = \text{Mean of Distrbution}$$
$$ x_i = \text{The ith value of x}$$
$$ n = \text{Number of Data Points} $$
$$\sigma = \text{Population Standard Deviation}$$
$$\overline{x} = \text{Mean of Distrbution}$$
$$ x_i = \text{The ith value of x}$$
$$ n = \text{Number of Data Points} $$
Equation
$$s = \sqrt{\frac{\sum_{i=1}^{n}(x_i-\overline{x})^2}{n-1}}$$
$$s = \text{Sample Standard Deviation}$$
$$\overline{x} = \text{Mean of Distrbution}$$
$$ x_i = \text{The ith value of x}$$
$$ n = \text{Number of Data Points} $$
$$s = \text{Sample Standard Deviation}$$
$$\overline{x} = \text{Mean of Distrbution}$$
$$ x_i = \text{The ith value of x}$$
$$ n = \text{Number of Data Points} $$
Challenge
Manually calculate the mean and both standard deviations.