Describing Data Numerically

Measures of Center and Spread in Symmetric Distributions

When analyzing numerical data, it is essential to summarize the distribution using measures of center and spread. For symmetric distributions, the mean is the primary measure of center, while the standard deviation quantifies the horizontal spread or variability of the data.

The Mean

Definition and Interpretation

The mean (often denoted as x̄, pronounced "x-bar") is the arithmetic average of a set of values. It can be thought of as the "balancing point" of the distribution, representing a typical value when the data are symmetric.

• Formula: The mean of a sample of size n with values x_1, x_2, ..., x_n is given by:

• Interpretation: The mean is a good estimate of a typical value for symmetric distributions, but may not represent a typical value in skewed distributions.

Mean in Symmetric vs. Skewed Distributions

For symmetric distributions, the mean accurately reflects the center of the data. For skewed distributions, the mean can be misleading, as it is influenced by extreme values (outliers).

• Symmetric Distribution: Mean is a good measure of center.

• Skewed Distribution: Mean may not represent a typical value; median is often preferred.

Calculating the Mean: Example

Suppose we have the following gas prices (in dollars) from 10 gas stations: 3.19, 3.09, 3.09, 2.93, 2.95, 3.09, 2.99, 2.99, 2.95, 2.97. The mean is calculated as:

• Sum of values:

• Number of observations:

• Mean:

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

Using Technology to Calculate the Mean

For large data sets, calculators or statistical software are used to compute the mean. It is important to understand how to interpret the results, not just how to compute them.

Measuring the Spread: Standard Deviation

Definition and Importance

The standard deviation measures how far the typical observation is from the mean. It is a key measure of variability in a data set, especially for symmetric distributions.

• Formula for Sample Standard Deviation:

• Interpretation: Most data values lie within one standard deviation of the mean in a symmetric distribution.

Example: Comparing Spread in Two Cities

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, indicating greater variability.

Standard Deviation: Smog Levels Example

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 particulate levels within one standard deviation of the mean (between 5.7 and 10.1), which is considered safe by EPA standards.

Standard Deviation: Basketball Team Heights Example

Consider the heights of players on two basketball teams:

• Team I: Standard deviation ≈ 2.4 inches

• Team II: Standard deviation ≈ 6.2 inches

• Interpretation: Team II has more variation in player heights than Team I.

Standard Deviation: Gas Prices Example

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

Variance

The variance is the square of the standard deviation. It is less commonly used because its units are the square of the original data's units, making interpretation less intuitive.

• Formula for Sample Variance:

• Relationship:

Summary Table: Mean and Standard Deviation