Measures of Center and Spread

Overview

Numerical summaries provide precise, quantitative descriptions of data distributions, complementing graphical displays. The two main concepts are center (location of the typical value) and spread (degree of variability). This unit focuses on the mean, median, standard deviation, interquartile range (IQR), range, and the use of z-scores and the Empirical Rule.

Measures of Variation

Key Definitions

• Range: The difference between the maximum and minimum values in a dataset. Formula:

• Sample Variance (): The average of the squared deviations from the mean, using in the denominator for samples. Formula:

• Sample Standard Deviation (): The square root of the variance, representing the typical distance that observations fall from the mean. Formula:

Using the TI-84 Calculator

1. Press [STAT], then select 1: Edit to access the data editor.

2. Enter your data into L1, clearing old data as needed.

3. Press [STAT], arrow to CALC, select 1: 1-Var Stats, and press ENTER.

4. Read the output:

   • : Sample mean

   • : Sample standard deviation (use for sample statistics)

   • : Population standard deviation

   • : Number of data points

5. To find variance, square the value:

Key Properties of Standard Deviation

• Units: Same as the original data.

• Non-negativity: Always .

• Zero Value: Equals zero only if all values are identical.

• Linear Shifts: Adding/subtracting a constant to all data points changes the mean but not the standard deviation.

• Outlier Sensitivity: Outliers increase standard deviation; removing them decreases it.

Examples

• Constant Shift Rule: Adding 15 years to each age in a dataset with years leaves unchanged.

• Removing Outliers: Removing a high home price ( thousand) reduces from to thousand.

• Zero Variability: If all values are the same, .

Measures of Center

The Mean as a Balancing Point

• Definition: The arithmetic average; the "balancing point" of a distribution.

• When to Use: For symmetric, unimodal distributions.

• Skewness Warning: In skewed distributions, the mean is pulled toward the tail and may not represent a typical value.

• Formula:

Example: Gas Prices

• Data: 12 gas prices, sum

• Mean:

• Interpretation: The typical price is

Using Technology for Large Datasets

• Statistical software (StatCrunch, Minitab, Excel, TI calculators) efficiently computes means for large datasets.

• Interpret output labels such as "Mean" or "Mean PM2.5DailyMean" as the average value.

Limitations of the Mean in Skewed Distributions

• Mean is sensitive to extreme values and may not reflect the typical value in skewed data.

• Example: NY State income: mean = , but median = (more representative).

The Median as the Middle Value

• Definition: The value that divides a sorted dataset into two equal halves.

• When to Use: For skewed distributions or when outliers are present.

• Calculation Steps:

  1. Sort the data.

  2. If is odd, median is the middle value.

  3. If is even, median is the average of the two middle values.

Example: NY vs. FL Median Income

• NY median: ; FL median:

• Interpretation: The typical NY resident earns more than the typical FL resident.

Measuring Variation: IQR and Range

Interquartile Range (IQR)

• Definition: The range of the middle 50% of data;

• Quartiles:

  • : Median of the lower half (25th percentile)

  • : Median (50th percentile)

  • : Median of the upper half (75th percentile)

• Resistance to Outliers: IQR is not affected by extreme values.

Example: Heights of 8 Children

• Data: 48, 48, 53, 53.5, 54, 60, 62, 71 (inches)

• Median (): inches

• : inches

• : inches

• IQR: inches

• Interpretation: The middle 50% of heights vary by 10.5 inches.

Range

• Definition:

• Comparison: Range is quick to compute but highly sensitive to outliers; IQR is more robust.

The Empirical Rule and Z-Scores

The Empirical Rule (68-95-99.7% Rule)

• Applies to: Unimodal, symmetric (bell-shaped) distributions.

• Percentages:

  • 68% of data within standard deviation ()

  • 95% within standard deviations ()

  • 99.7% within standard deviations ()

Z-Scores (Standard Scores)

• Definition: The number of standard deviations a value is from the mean.

• Formula:

• Interpretation:

  • : Value above the mean

  • : Value below the mean

  • : Value equals the mean

• Purpose: Allows comparison across different groups or units.

Example: Height Z-Score

• Value: 75 inches, Mean: 70 inches, SD: 3 inches

• Interpretation: 1.67 standard deviations above average

Identifying Unusual Values

• Values with are generally considered unusual or rare.

Finding Raw Values from Z-Scores

• Formula:

• Given a z-score, mean, and standard deviation, you can recover the original value.

Comparing Relative Standing Across Distributions

• Z-scores allow direct comparison of values from different populations or units.

• The value with the z-score furthest from 0 is more extreme relative to its group.

Summary Table: Measures of Center and Spread

Quick Reference: Key Formulas

• Sample Mean:

• Sample Variance:

• Sample Standard Deviation:

• Range:

• Interquartile Range:

• Z-Score:

• Raw Value from Z-Score:

Study Tip: Always write the z-score formula and the threshold at the top of your scratch paper for quick reference during exams.

Additional info: The notes above expand on the original content by providing full definitions, step-by-step calculation methods, and context for when to use each measure. Examples and formulas are included for clarity and exam preparation.