Gas Prices and Z-Scores Analysis

Analysis of Gas Prices Across Different States

  • Examining Gas Prices:

    • Purpose: Assess the variance of gas prices across multiple states to identify trends.

    • Sample Selection: 10 gas stations chosen from varying states to ensure a representative sample.

  • Data Recorded:

    • Average Price of Regular Gas:

    • Recorded Average: $3.31 per gallon.

    • California Gas Station:

    • Price: $4.66 per gallon.

    • Corresponding Z-Score: 2.04.

    • The z-score indicates how many standard deviations the California gas price is from the mean of the distribution of gas prices across the selected stations.

Calculation of Standard Deviation

  • Objective: Find the standard deviation of the gas prices for the 10 gas stations.

  • Formula for Z-Score:

    • z=(X−μ)σz = \frac{(X - \mu)}{\sigma}

    • Where:

      • zz = z-score

      • XX = value from the dataset (in this case, the gas price in California)

      • μ\mu = mean of the dataset (in this case, $3.31)

      • σ\sigma = standard deviation of the dataset

  • Rearranging the Formula:

    • To find σ\sigma, rearrange the equation:

    • σ=(X−μ)z\sigma = \frac{(X - \mu)}{z}

  • Inserting Values:

    • Plugging in the values:

    • X=4.66X = 4.66 (California gas price)

    • μ=3.31\mu = 3.31 (average price)

    • z=2.04z = 2.04

    • σ=(4.66−3.31)2.04\sigma = \frac{(4.66 - 3.31)}{2.04}

    • σ=1.352.04\sigma = \frac{1.35}{2.04}

    • σ≈0.6618\sigma \approx 0.6618

Comparison of Two Different Values from the Same Distribution

  • Values A and B:

    • Value A has a z-score of 1.4.

    • Value B has a z-score of -2.1.

a. Distance from the Mean

  • Definition of Distance from Mean:

    • The distance from the mean can be determined by the absolute value of the z-scores (ignoring the sign).

  • Calculating Absolute Values:

    • For Value A (z = 1.4): Absolute distance = 1.4.

    • For Value B (z = -2.1): Absolute distance = 2.1

  • Conclusion:

    • Value B is farther from the mean since its absolute z-score (2.1) is greater than that of Value A (1.4).

b. Percentile Comparison

  • Understanding Percentiles:

    • A higher z-score indicates a higher percentile rank, meaning Value A has more data points below it compared to Value B.

  • Conclusion:

    • Value A with a z-score of 1.4 is at a higher percentile than Value B with a z-score of -2.1.

    • This reinforces that Value A is closer to the mean, thereby positioned within a higher ranking in the overall distribution of values.