Describing Data Numerically

Measures of Center and Spread in Symmetric Distributions

When analyzing numerical data, it is essential to describe the distribution using measures of center and spread. For symmetric distributions, the mean is the primary measure of center, and the standard deviation is the main measure of spread.

• Center (Balance Point): The mean represents the balancing point of the distribution.

• Spread (Variability): The standard deviation quantifies how much the data values deviate from the mean.

The Mean

The mean (often called the arithmetic average) is a measure of center that can be thought of as the balancing point of the distribution. It is especially useful for symmetric distributions, where it accurately represents a typical value in the data set.

• Definition: The mean is the sum of all data values divided by the number of observations.

• Symbol: The sample mean is denoted as (x-bar).

• Formula:

• Interpretation: For symmetric distributions, the mean is a good estimate of a typical value.

Example: Gas Prices

Suppose a sample of prices for 1 gallon of regular gas at 10 different gas stations in Austin, Texas, is collected:

• Data: $3.19, $3.09, $3.09, $2.93, $2.95, $3.09, $2.99, $2.99, $2.95, $2.97

• The mean price is $3.02.

• Interpretation: The typical price of 1 gallon of gas at these stations was $3.02 on that day.

The Mean in Skewed Distributions

For skewed distributions, the mean may not represent a typical value well, as it is influenced by extreme values (outliers).

• In skewed data, the mean can be pulled toward the tail, making it less representative of most data points.

• Example: In tennis player winnings, the mean is much higher than what most players actually earned.

Calculating the Mean

For small data sets, calculate the mean by hand using the formula above. For large data sets, use statistical software or calculators. Interpretation and understanding of the mean are more important than manual calculation in most practical scenarios.

Measuring the Spread: Standard Deviation

Understanding Spread

The spread of a distribution describes how much the data values vary around the center. The standard deviation is the most common measure of spread for symmetric distributions.

• Definition: The standard deviation measures the typical distance of the data values from the mean.

• For most distributions, a majority of the data is within one standard deviation of the mean.

Example: Comparing Temperature Distributions

Consider the daily high temperatures in Provo, Utah, and San Francisco, CA. Both distributions are symmetric, but Provo's temperatures are more spread out than San Francisco's.

Standard Deviation: Calculation and Interpretation

• Formula:

• Interpretation: The standard deviation tells us how much the data typically varies from the mean.

Example: Smog Levels

The distribution of particulate matter in 371 U.S. cities has a mean of 7.9 and a standard deviation of 2.18 micrograms per cubic meter. Most cities have levels between one standard deviation above and below the mean (5.7 to 10.1), which is considered safe by the EPA.

Example: Basketball Player Heights

Comparing the heights of players on two basketball teams, Team II has a larger standard deviation, indicating more variability in player heights compared to Team I.

Example: Gas Prices (Standard Deviation)

For the gas price data, the standard deviation is about $0.09, meaning most prices are within 9 cents of the mean price of $3.02.

Variance

The variance is the square of the standard deviation and is less commonly used because it is in squared units. The standard deviation is preferred for interpretation since it is in the same units as the original data.

Summary Table: Mean and Standard Deviation

Using Technology

Statistical calculators and software can quickly compute the mean and standard deviation. However, it is crucial to understand what these statistics represent and how to interpret them in context.