Chapter 3: Data Description

Measures of Central Tendency

Parameters vs Statistics

Parameters: taken from the entire population

Statistics: taken from a sample

Mean

Mean: Arithmetic average

Notations:

$\mu$ for population
$\overline{X}$ for sample

Formula:

\[\mu =\frac {\Sigma X} {n}\]

where

$X$ is the data; and
$n$ is the number of values

Mean for Weighted Data

Calculation:

To find $\Sigma X$, sum the product of each value and its corresponding weight
To find $n$, sum the weights

Formula:

\[\frac {\Sigma (X \cdot w)} {\Sigma w}\]

where

$X$ is the data; and
$w$ are the weights

Mean for Grouped Data

Calculation: To find $\Sigma X$, sum the midpoints of each class, weighted by the class frequency

Formula:

\[\frac {\Sigma (f \cdot X_m)} {n}\]

where

$f$ is the class frequency;
$X_m$ is the class midpoint; and
$n$ is the number of values

Median

Median (MD): Value in the middle of the data set.

Where n is odd, the middle value
Where n is even, the average of the two middle values

Median of grouped data

Find the group with the median value
Find the ratio of how many values into the group the median is ($\frac {value \space index}{class \space frequency}$)
Apply the ratio to the class width; add to class lower boundary

Example:

$n$: 61
Class with $i_{31}$ contains the median
Class “15-20” contains $i_{30}-i_{36}$
$i_{31}$ is 2 values into the class
The class frequency is 7
The ratio of the values into the class the median is is $\frac {2} {7}$
The class width is 5
The applied ratio is $\frac {10}{7}$
The lower boundary is 14.5
With the added $\frac{10}{7}$ we round to 16
Median is: 16

Mode

Mode: most often occurring value

Data can be unimodal, bimodal, multimodal, or have no mode.
For grouped data, the class with the highest frequency is the modal class.

Midrange

Midrange (MR): average of min and max

Outliers

Outliers: Extremely high or low values

Distribution Shapes

Symmetric distribution:
- Data is evenly distributed on both sides of the mean
Right skewed /positively skewed:
- Most of the data is on the left side of the mean; the mean is pulled right due to right outliers
- The median is to the left of the mean
- Generally, the mode is to the left of the mean and the median
Left skewed / negatively skewed:
- Most of the data is on the right side of the mean; the mean is pulled left due to left outliers
- The median is to the right of the mean
- Generally, the mode is to the right of the mean and the median

Measures of Variation

Range

Range (R): Distance between the highest and lowest values

Formula:

\[X_{max}-X_{min}\]

Variance, Standard Deviation

Deviation: A value’s deviation is its distance from the mean

Population Variance $(\sigma ^2)$:

Average squared data point deviation for the population data
Measures the distribution spread
Formula:

\[{ \frac{\Sigma(X-\mu)^2}{n} }\]

where:

$X$ is the individual data;
$\mu$ is the population mean; and
$n$ is the number of values

Population Standard Deviation $(\sigma)$:

Quadratic mean of the deviation of population data
Square root of the population variance
Formula:

\[\sqrt { \frac{\Sigma(X-\mu)^2}{n} }\]

Sampled Data Standard Deviation $(s)$ and Variance $(s^2)$:

$(n - 1)$ is used in the denominator instead of $n$
Formula:

\[\sqrt { \frac{\Sigma(X-\overline{X})^2}{n-1} }\]

where:

$X$ is the individual data;
$\overline{X}$ is the population mean; and
$n$ is the number of values

There is a shortcut formula which does not require a precalculated mean:

\[\sqrt{ \frac{n(\Sigma X^2)-(\Sigma X)^2}{n(n-1)} }\]

Grouped Data Standard Deviation: Class midpoint’s deviation, weighted by class frequency

Shortcut formula:

\[\sqrt{ \frac{n(\Sigma f \cdot X_{m}^2)-(\Sigma f \cdot X_m)^2}{n(n-1)} }\]

Coefficient of Variation

Coefficient of Variation: Standard for variance measurement; ratio of standard deviation to mean

For populations: $(\frac{\sigma}{\mu}\cdot 100) \%$
For samples: $(\frac{s}{\overline{X}}\cdot 100) \%$

Distribution Rules

Range rule of thumb: The standard deviation is roughly a quarter of the range

Formula:

\[s \approx \frac {range}{4}\]

Chebyshev’s Theorem: finds percentage of values within $x$ standard deviations from the mean; applicable to all distribution shapes:

Theorem: Within $k$ standard deviations from the mean will be at least $(1- \frac{1}{k^2})\%$ of the values, where $k$ is greater than 1.

$75\%<2\sigma$
$88.9\%<3\sigma$

Empirical Rule: finds percentage of values within $x$ standard deviations from the mean in a normal / bell-shaped distribution:

$68\%<1\sigma$
$95\%<2\sigma$
$99.7\%<3\sigma$

Measures of Position

Z Score

Z score: Value’s standard-deviation units from the mean

Calculation: Distance from the mean divided by the standard deviation

Formula:

\[\frac{X-\overline{X}}{S};\frac{X-\mu}{\sigma}\]

Percentiles

Percentiles divide data into 100 groups

1% of data is below $P_1$, 20% below $P_{20}$ etc.

Calculation: To find the percentile of $X$ within a dataset with $n$ values, divide $X$’s cumulative frequency by $n$

Formula:

\[(\frac{X_{CF}}{n}\cdot100)\%\]

Tweaked Formula: To obtain a more representative view, we use a tweaked formula:

To find the percentile of a value:

Calculation: The number of values less than $X$ plus 0.5, over $n$

Formula:

\[\frac { n _{ \< x} + 0.5 } {n} \cdot 100 \%\]

To find the value corresponding to a percentile:
- First, find $\frac{n\cdot P}{100}$
- If $\frac{n\cdot P}{100}$ is a fraction, round up and find the correspondingly ranked data
- Formula:

\[X_{\lceil \frac{n\cdot P}{100} \rceil}\]

If $\frac{n\cdot P}{100}$ is a whole number, use the average of the value with that rank and the next value
Formula:

\[\text{average}({X_{ \frac{n \cdot P}{100} }, \space X_{ \frac{n \cdot P}{100} +1}})\]

Quartiles

Quartiles divide data into 4 groups:

$Q_1=P_{25}$
$Q_2=P_{50}$
$Q_3=P_{75}$

For a more direct quartile formula:

$Q_2$: Median of all values
$Q_1$: Median of values below $Q_2$
$Q_3$: Median of values above $Q_2$

Interquartile Range

Interquartile range (IQR): $Q3-Q1$

Procedure to identify outliers: $Q1-(IRQ \cdot 1.5)$ or $Q3+(IRQ \cdot 1.5)$

Decile

Decile divides data into 10 groups; $D_1=P_{10}$…

Exploratory Data Analysis

Five number summary: Includes

Min
Q1
Median
Q3
Max

Boxplot: Graph of five number summary: Vertical line at median; box extending to Q1 and Q3; horizontal line in middle of box to min and max.

Example:

$\min=3$
$Q_1=9$
$\text{med}=18$
$Q_3=21$
$\max = 27$