Basics of Exploring Numerical Data

Variables

Types of Data: Continuous vs Discrete
Descriptive Statistics: Measures of center (mean, median, mode) and spread (range, variance, standard deviation, IQR)

Visualization Techniques

Dotplots: Understanding frequency distributions
Histograms: Viewing shapes of distributions
Boxplots: Detecting outliers and spread
Scatterplots: Identifying relationships between variables

Scatterplots

Scatterplots are useful for visualizing the relationship between two numerical variables.

Example:

The variables Sepal.Length and Petal.Length in the iris data set appears to have a positive association, meaning as Sepal.Length increases, Petal.Length also increases.

Dot Plots

Dot plots are useful for visualizing one numerical variable. Darker colors represent areas where there are more observations.

Dot Plots and the Mean

The mean, also called the average (marked with a triangle in the above plot), is one way to measure the center of a distribution of data. The mean GPA is 3.59.

Histograms

Histograms provide a view of the data density. Higher bars represent where the data are relatively more common.

Histograms are especially convenient for describing the shape of the data distribution.
The chosen bin width or number of bins can alter the story the histogram is telling.

Bin Width of Histograms

Which one(s) of these histograms are useful? Which reveal too much about the data? Which hide too much?

Distribution Shapes: Modality

Does the histogram have a single prominent peak (unimodal), several prominent peaks (bimodal/multimodal), or no apparent peaks (uniform)?

$\star$ In order to determine modality, step back and imagine a smooth curve over the histogram –imagine that the bars are wooden blocks and you drop a limp spaghetti over them, the shape the spaghetti would take could be viewed as a smooth curve.

Distribution Shapes: Skewness

Is the histogram right skewed, left skewed, or symmetric?

$\star$ Histograms are said to be skewed to the side of the long tail.

Commonly Observed Distribution Shapes

Measures of Central Tendency

The measures of central tendency describe the central or typical value of a dataset, summarizing its distribution. The following are the common measures of central tendency:

Mean: The arithmetic average of all data points, calculated as the sum of all values divided by the total number of values.
Median: The middle value of an ordered dataset. If there is an even number of observations, the median is the average of the two middle values.
Mode: The most frequently occurring value(s) in a dataset. A dataset may have one mode (unimodal), multiple modes (multimodal), or no mode if all values occur with equal frequency.

Skewness and Measures Central Tendency (1/3)

Mode $<$ Median $<$ Mean

If the distribution of data is skewed to the right, the mode is less than the median, which is less than the mean.

Skewness and Measures Central Tendency (2/3)

Mean $<$ Median $<$ Mode

If the distribution of data is skewed to the left, the mean is less than the median, which is less than the mode.

Skewness and Measures Central Tendency (3/3)

Mean $=$ Median $=$ Mode

If the distribution of data is symmetric, the mean is roughly equal to the median, which is roughly equal to the mode.

Measures of Dispersion (Spread)

The measures of dispersion describes the variability of a numerical data. It is a way to quantify the uncertainty of a distribution. The following are the common measures of dispersion:

Variance: The average squared deviation from the mean.
Standard Deviation: The square root of the variance, providing a measure of spread in the same units as the data.
Range: The difference between the maximum and minimum values.
Interquartile Range (IQR): The difference between the 75th and 25th percentiles, capturing the spread of the middle 50% of the data.

Making Sense of the Variance

Common variance interpretations:

Zero Variance: All values are the same.
Low Variance: Data points are close to the mean (consistent data).
High Variance: Data points are spread out (greater variability).

Example: Order the following distributions from low to high variance.

The above distributions are ordered from lowest to highest variance as follows: B, D, C, and A.

Making Sense of the Standard Deviation

Variance gives a broader measure of spread, while standard deviation provides a more practical understanding of dispersion.

Example: Suppose we analyze the annual salaries of employees in two different companies:

Company	$\overline{x}$	$s^2$	$s$
A	$55K	$62.5K	$7.906K
B	$130K	$2250K	$47.434K

Even though Company B has higher salaries on average, its standard deviation is much larger, suggesting greater salary inequality.

$\star$ Standard deviation helps compare the spread of different distributions of the same units, which is directly interpretable as the typical deviation from the mean.

Box Plots

The box in a box plot represents the middle 50% of the data, and the thick line in the box is the median.

Box Plot of Study Hours

Histogram of Study Hours

$\star$ Box plots and histograms visualize numerical data, with histograms showing distribution shape, while box plots summarize spread and outliers, making them better for comparisons.

Anatomy of Box Plots

Main Parts:

Box
- Defined by the quartiles $Q_1$, $Q_2$ (median), and $Q_3$.
- The IQR defines the length of the Box.
Whiskers
- Lower whisker is defined as $Q_1 - 1.5 \times \text{IQR}$.
- Upper whisker is defined as $Q_3 + 1.5 \times \text{IQR}$.
Outliers
- Data points that are placed beyond the whiskers.

Computing the Whiskers

Whiskers of a box plot can extend up to $1.5 \times IQR$ away from the quartiles. The $1.5 \times \text{IQR}$ is arbitrary, and is considered an academic standard and the default in plotting box plots.

Example: Suppose that $Q_1 = 10$ and $Q_3 = 20$.

\[\text{IQR} = 20 - 10 = 10\] \[\text{max upper whisker reach} = 20 + 1.5 \times 10 = 35\] \[\text{max lower whisker reach} = 10 - 1.5 \times 10 = -5\]

$\star$ A potential outlier is defined as an observation beyond the maximum reach of the whiskers. It is an observation that appears extreme relative to the rest of the data.

More Boxplot and Dot Plot Examples

Example: The boxplot and stacked dot plot of the data set $7,1,2,4,6,3,2,7$ is shown below.

Box Plot

Stacked Dot Plot

$\star$ A dot plot is used instead of a histogram for convenience with the small discrete numbered dataset. The box plot shows no potential outliers, as all data points fall within the whiskers.

Why Outliers are Important?

Identify extreme skew in the distribution.
Identify data collection and entry errors.
Provide insight into interesting features of the data.

Case Study 1

How would sample statistics such as mean, median, SD, and IQR of household income be affected if the largest value was replaced with $10 million? What if the smallest value was replaced with $10 million?

Case Study 1: Extreme Observations

data	$\overline{x}$	$s$	Median	IQR
original	244,778.6	225,903.8	190,000	200,000
replace largest with $10M	566,207.1	1,830,200.6	190,000	200,000
replace smallest with $10M	316,121.4	854,489.2	200,000	200,000

$\star$ The table shows that shifting specific values to the extreme significantly affects the mean but not the median, indicating the mean’s sensitivity to extreme observations. Similarly, the standard deviation is affected, while the IQR remains the same.

Robust Statistics

Median and IQR are more robust to skewness and outliers than mean and SD. Therefore,

for skewed distributions it is often more helpful to use median and IQR to describe the center and spread
for symmetric distributions it is often more helpful to use the mean and SD to describe the center and spread

Exploring Numerical Data

Applied Statistics

Objectives