Mean

The sample mean of a numerical variable is computed as the sum of all of the observations divided by the number of observations:

\(\bar x = \sum_{i=1}^n x_i = \frac{(x_1+x_2+...+x_n)}{n}\)

Median: the number in the middle

The median splits an ordered data set in half. If there are an even number of observations, the median is the average of the two middle values. If there are an odd number of observations, the median is the middle value.

 [1]  0  0  0  0  0  0  1  1  1  1  1  2  2  3  3  3  4  4  5  5  5  6  6
[24]  7  7  7  9  9  9 10 10 10 11 11 12 14 14 16 17 22 25 25 25 26 26 27
[47] 29 42 43 64

\(n\) is odd

1. Sort the series in ascending order.
2. If the series has odd number \((n)\) of entries, the median is at position \(\frac{n+1}{2}.\)
3. Find the median of the series: \(2,4,5,(6),7,9,9\)
4. The median is \(6.\)

\(n\) is even

1. Sort the series in ascending order.
2. If the series has even number \((n)\) of entries, the median is the average of the two middle numbers: \(\frac{n}{2},\frac{n+1}{2}.\)
3. Find the median of the numbers: \(2,2,4,6,7,8\)
4. Median is the average of the third and the fourth numbers: \(\frac{4+6}{2}=5\)

The weighted mean is the same as the mean, except that it is influenced more by some observations than others. We assign weights to observations as a sort of way of describing its relative importance.

The weighted mean of observations \(x_1, x_2,...,x_n\) using weights \(w_1, w_2,...,w_n\) is given by

\(\bar x =\frac{w_1x_1+w_2x_2+...+w_nx_n}{w_1+w_2+...+w_n}\)

\(\bar x =\frac{1\times x_1+1\times x_2+...+1\times x_n}{1+1+...+1} = \frac{x_1+x_2+...+x_n}{n}\)

The midrange of a data set is the measure of center that is the value midway between the maximum and minimum values in the original data set. It is found by adding the maximum data value to the minimum data value and then dividing the sum by \(2\), as the following formula:

\[ Midrange = \frac{\text{maximum data value + minimum data value}}{2} \]

The range of a set of data is the difference between the maximum and the minimum data values.

\(range = maximum - minimum\)

The range is sensitive to outliers. A single high or low value will affect the range significantly.

• Percentiles divide the data in one hundred groups.
• The \(n^{th}\) percentile is the data value such that \(n^{th}\) percent of the data lies below that value.

\[ \text{Percentile of value x} = \frac{\text{number of values less than x}}{\text{total number of values}} \times 100 \]

Three Quartiles \((Q_1, Q_2, Q_3)\)

• \(Q_1\) represents the first quartile, which is the 25th percentile, and is the median of the smaller half of the data set.
• \(Q_2\) represents the second quartile, which is equivalent to the 50th percentile (i.e. the median).
• \(Q_3\) represents the third quartile, or 75th percentile, and is the median of the larger half of the data set
  
• Interquartile Range \((IQR) = Q_3 - Q_1\)

Outliers in the context of a box plot

When in the context of a box plot, define an outlier as an observation that is more than \(1.5 \times IQR\) above \(Q_3\) or \(1.5 \times IQR\) below \(Q_1\). Such points are marked using a dot or asterisk in a box plot.

Data: \([5, 5, 9, 10, 15, 16, 20, 30, 40]\)

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
   5.00    9.00   15.00   16.67   20.00   40.00 

• SD \((s)\) of a set of sample values is a measure of how much, on average, the data values deviate away from the sample mean.
• In other word, SD describes the variability of the data set within the range of the dataset.
  • Low variability or small spread means that the values tend to be more clustered together.
  • High variability or large spread means that the values tend to be far apart.

Calculating the Standard Deviation

The standard deviation is the square root of the variance. It is roughly the average distance of the observations from the mean.

\[ \bbox[yellow,5px] { \color{black}{s= \sqrt{\frac{1}{n-1}\sum(x_i-\bar x)^2}} } \]

\(Calculate \space SD \space of \space [0,1]\)

Notice the spread of the distributions.

\[ \bbox[yellow,5px] { \color{black}{\sigma= \sqrt{\frac{1}{N}\sum(x_i-\mu)^2}} } \] Variance of a Sample and Population

The variance of a set of values is a measure of variation equal to the square of the standard variation.

• Sample variance: \(s^2 =\) square of the sample standard deviation \(s\).
• Population variance: \(\sigma^2 =\) square of the population standard deviation \(\sigma\).

Coefficient of Variation

The coefficient of variation (CV) for a set of nonnegative sample or population data, expressed as a percent, describes the standard deviation relative to the mean, and is given by the following:

\[ Sample: CV = \frac{s}{x}.100 \\ Population: CV = \frac{\sigma}{x}.100 \]

Probabilities for falling 1, 2, and 3 standard deviations of the mean in a normal distribution.

Consider a normally distributed random variable \(x\) with mean \(\mu\) and sd \(\sigma\): \(x \tilde \space N(\mu, \sigma)\)

Two-step linear transformation of \(x\)

1. subtract \(\mu\) from \(x\)
   
2. divide \((x-\mu)\) by \(\sigma\)

\[\bbox[yellow,5px]{\color{black}{\text{standard normal deviate: } z = \frac {x-\mu}{\sigma}}}\]

The Z-score of an observation is defined as the number of standard deviations it falls above or bemow the mean. If the observation is one standard deviation above the mean, its Z-score is 1. If it is 1.5 standard deviations below the mean, then its Z-score is -1.5.

The normal distribution model describes a symmetric, unimodal, bell-shaped curve. It can be adjusted using two parameters; mean \((\mu)\) and standard deviation \((\sigma)\).

\[ \bbox[yellow,5px] { \color{black}{{\text {Density at z}} = \frac {1}{\sqrt {2\pi}}\exp{-\frac{1}{2}z^2}, -\infty<z<+\infty} } \]

Comparing distributions of median household income for counties by population gain status

^{Source: OpenIntroOrg}

• Data with outliers (i.e. skewed distribution) are hard to interpret
• Such data are often transformed into their logarithmic form to give the skewed distribution a normal (i.e. roughly unimodal and symmetric) shape for better interpretation