Lecture_4

extra office hour from undergraduates next week.

Averages

• let \(v_i\), … \(v_n\) to denote the valued in population of our variable of interest. The population average or population mean is

\[ \mu = {1\over{N}}\sum^n_{i=1}{v_i} \]

you may keep two digit for accuracy in homework.

• sample mean (or sample average)

\[ \bar{x} = {1\over{n}}\sum^n_{i=1}{x_i} \]

Variance and standard deviation

• The population variance measures the amount of intrinsic variability in the population (average distance from the population mean)

\[ \sigma^2={1\over{N}}\sum^N_{i=1}{(v_i-\mu)^2} \]

The square function is simpler than absolute value (square function is differentiable and looks nice.)

• We can also demote the population variance as \(\sigma^2_X\) or Var(X), the variance of the random variable X.

• The population standard deviation is the square root of the population variance.

\[ \sigma = \sqrt{\sigma^2} \]

unit of variance: (original unit)^2
unit of standard deviation: original unit

• The sample variance \[ S^2={1\over{n-1}}\sum^n_{i=1}{(x_i-\bar{x})^2} \]

we use n-1 to make estimation unbiaseness of S^2 for ^2. (we don’t have full scope for the data anymore when we use sample to esitmate population)

• The sample standard deviation \[ S=\sqrt{S^2} \]

if all the sample response are the same, S=0. (no variability in data)

In R

Sample data

temperature<-c(1,6,3,4,5,2,7)
mean(temperature)

## [1] 4

var(temperature)

## [1] 4.666667

sd(temperature)

## [1] 2.160247

Percentile

The \(x_{(i)}\) is the 100(\({i-0.5}\over n\))-th sample percentile. The 0.5 here is used to avoid 100 percentile.

In R

sort(temperature)

## [1] 1 2 3 4 5 6 7

Some percentiles hare of particular interest:

• 50th percentile = median = \(\widetilde{x}\)
• 25th percentile = lower sample quantile = \(q_1\)
• 75th percentile = higher sample quantile = \(q_3\)

In R

quantile(temperature,0.25)

## 25% 
## 2.5

quantile(temperature,0.75)

## 75% 
## 5.5

# median
quantile(temperature,0.5)

## 50% 
##   4

median(temperature)

## [1] 4

• The R command summary provides five number summary for the data \(min,q1,median,q2,max\)

summary(temperature)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##     1.0     2.5     4.0     4.0     5.5     7.0

sample average is sensitive for outliers, while the median is less sensitive to outliers.

The sample interquartile range (IQR) is an alternative measure of variability. \[ IQR=q_3-q_1 \] Boxplots

\[ max=min(q_3+1.5\times IQR,actual\ max) \\ min=max(q_1-1.5\times IQR,actual\ min) \] exceptions are called outliers.