Sept 1/2

Sept 1,2 – Statistics

Chapter 2 – numeric measures of data sets.

Measures of central tendency

Median, mean, mode

Mean (average): $\bar{x}=\frac{\sum\limits_{i=1}^{n}{{{x}_{i}}}}{n}$

Median: middle (from bottom to top); position (n+1)/2

Mean is dependent on values; median is resistant to the change of the values at the top or the bottom.

When the data set is placed in the TI, the mean and median are computed by doing 1-var stats.

To enter the data, do STATS EDIT 1.

To compute the values, do STATS CALC 1-Var Stats.

Quartiles: dividing points for the division of the data into quarters.

The TI computes the quartiles automatically. You must scroll down to see them.

The five-number summary is the set of low, Q1, median, Q3, and high.

Q1 is the median of the lower half; Q3 is the median of the upper half.

This is plotted as a box and whisker plot on the TI.

You do this, once the data is entered, by doing STAT PLOT and describing the data location and the type of display required.

The mode is the most common data value. Some data sets have no mode; some have only one, some have several modes. This is a less commonly used measure of central tendency.

How much does the data spread out?

First: range. This is simply maximum – minimum.

Second: Interquartile range: Q3-Q1

The third measure:

The average square variation from the mean is the VARIANCE.

We subtract the mean from each data value, square each difference, and average these quantities:

${{s}^{2}}=\frac{\sum\limits_{i=1}^{n}{{{({{x}_{i}}-\bar{x})}^{2}}}}{n-1}$.

And the square root of this is the standard deviation.

$s=\sqrt{\frac{\sum\limits_{i=1}^{n}{{{({{x}_{i}}-\bar{x})}^{2}}}}{n-1}}$

The variance is always positive, as every value is squared. The number of degrees of freedom is n-1.

The units for the standard deviation are the same as for the data.

The most important question is how to organize and study the data. This is the content of section 2-10. Making these choices depends on a careful analysis of the problem, before any computations are completed.