• Shape: modality, skewness
• Center: mean, median, mode
• Spread: variance, sd, range, IQR
• Unusual observations: outliers

Modality

Skewness

X <- c(8, 11, 7, 7, 8, 11, 9, 6, 10, 7, 9)

\[ \frac{8 + 11 + 7 + 7 + 8 + 11 + 9 + 6 + 10 + 7 + 9}{11} = \frac{93}{11} = 8.45 \\ \\ \]

Sample mean: the arithmetic mean of the data (vs pop mean)

\[ \bar{x} = \frac{x_1 + x_2 + \ldots + x_n}{n} \quad \quad \textrm{vs.} \quad \quad \mu\]

mean(X)

## [1] 8.5

Median: the middle value of a sorted data set.

sort(X)

##  [1]  6  7  7  7  8  8  9  9 10 11 11

median(X)

## [1] 8

Break ties by averaging middle two if necessary.

Mode: the most frequently observed value in the data set.

table(X)

## X
##  6  7  8  9 10 11 
##  1  3  2  2  1  2

Sample variance: roughly, the mean squared deviation from the mean.

\[ s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n - 1}\]

Compare to the population variance, \(\sigma^2\), which divides by \(n\).

X - mean(X)

##  [1] -0.45  2.55 -1.45 -1.45 -0.45  2.55  0.55 -2.45  1.55 -1.45  0.55

(X - mean(X))^2

##  [1] 0.21 6.48 2.12 2.12 0.21 6.48 0.30 6.02 2.39 2.12 0.30

sum((X - mean(X))^2) / (length(X) - 1)

## [1] 2.9

var(X)

## [1] 2.9

Sample standard deviation: the square root of the variance. Used because units are the same as the data.

\[ s = \sqrt(s^2) \]

sqrt(var(X))

## [1] 1.7

sd(X)

## [1] 1.7

Compared to the population standard deviation, \(\sigma\).

Inner Quartile Range: the range of the middle 50% of the data.

\[ IQR = Q3 - Q1 \]

sort(X)

##  [1]  6  7  7  7  8  8  9  9 10 11 11

IQR(X)

## [1] 2.5

Range: the range of the full data set.

\[ range = max - min \]

max(X) - min(X)

## [1] 5

range(X)

## [1]  6 11

X

##  [1]  8 11  7  7  8 11  9  6 10  7  9

Y

##  [1] 37 11  7  7  8 11  9  6 10  7  9

var(X)

## [1] 2.9

var(Y)

## [1] 77

IQR(X)

## [1] 2.5

IQR(Y)

## [1] 3.5

range(X)

## [1]  6 11

range(Y)

## [1]  6 37

The mean (red line) is sensitive to extreme values, so it gets pulled towards the tail. The median (dashed line) is less sensitive.

For symmetric dists, use mean and sd.

For skewed dists, use median and iqr.