Introduction to Statistics

• Basic Concept
• Exploring single variable
• Exploring multiple variable
• Advanced plots

• All statistical models are of the form \[ outcome_{i} = (Model)+error_{i} \]
• Mean is the simplest model (or a default model) if there is nothing
• A deviation is the difference beween the mean and the actual data point
• Deviation is also called Error
• In a more general sense, error is the difference between the model value and the actual data point

• Sum of squared errors (SSE) assesses the magnitude of total error
• Its value is dependent on the number of points
• It increases with the increase in number of points
• Mean Square Error (MSE) takes the average of SSE
• MSE= SSE/N-1 (in case of sample data)
• MSE is the same as variance

• The sum of squares, variance, and standard deviation represent the same thing:
  • The 'fit' of the mean to the data
  • The variability in the data
  • How well the mean represents the observed data
  • Error

• A statistic for which the frequency of particular values is known.
• Observed values can be used to test hypotheses. \[ test~statistic=\frac {\text {variance explained by the model}}{\text{variance NOT explained by the model}} \] \[ test~statistic=\frac {\text {effect}}{\text{error}} \]

• Standard error is the standard deviation of a test statistic
• Suppose we are trying to make an inference about the average weight of people in a city
• Suppose we take a random sample of 10 people and compute the mean
• If we repeat this process 500 times (say), then we can create a histogram of the 500 sample means
• The grand mean of all 500 sample means will approach the actual population mean (not observed)
• The standard deviation of the 500 sample means is called the Standard Error of the Mean (SME) denoted by \( \sigma_{\bar{X}} \)

• In practice, we are not going to draw 500 different samples (with each of size 10)
• When the sample size is bigger than 30, CLT says the distribution of sample means approximates Normal distribution
• So just one sample of size 10 is drawn
• SME gives the indication about the “unreliability” or spread of the grand mean from the actual population mean
• A small SE indicates that most sample means are similar to the population mean and so our sample is likely to be representative of the population

• \( SME=\sigma_{\bar{X}} = \sigma_{population}/\sqrt{N} \) where N is the sample size
• However, \( \sigma_{population} \) is not available
• But it can be approximated with standard deviation of the samples (10 numbers in our example) when the sample size is big (>30)
• \( SME=\sigma_{\bar{X}} = s/\sqrt{N} \)

• When the sample size is small (<30), the test statistic follows t-distribution
• For larger samples, it approaches normal distribution
• t-distribution incorporates more uncertainty (and hence wider Confidence Interval) than Normal distribution

sampleSize=21
data1=c(18,16,18,24,23,22,22,23,26,29,32,34,34,36,36,43,42,49,46,46,57)
var=var(data1); var

[1] 134.2619

s=sqrt(var); s

[1] 11.58714

mean=mean(data1); mean

[1] 32.19048

SSE=s/sqrt(sampleSize); SSE

[1] 2.528522

#95% CI using t-distribution because of small sample size
lower.boundary=mean-2.09*SSE
upper.boundary=mean+2.09*SSE
lower.boundary;upper.boundary

[1] 26.90586

[1] 37.47509