Module 7 – Z-Scores

Z-Scores

Introduction

• The lecture introduces the concept of Z-scores and how they are used to describe individual scores within a sample in terms of standard deviations from the mean.

• The initial question involves comparing two students, Peter and Mary, from different statistics courses to determine who performed better relative to their class.

• Peter scored 95 in professor 1's class where the mean was 85.

• Mary scored 90 in professor 2's class where the mean was 80.

• The answer to who performed better cannot be determined without knowing the variability (standard deviation) of scores in each class.

Understanding Variability

• If the standard deviation on professor 1’s test was 10, Peter is within 1 standard deviation from the mean, meaning his score isn't exceptionally high.

• If the standard deviation on professor 1’s test was 1, Peter’s score is 10 standard deviations from the mean, which is considered an outlier.

• A score more than 2.5 standard deviations from the mean is generally considered an outlier.

Review of Distribution Concepts

• The lecture reviews concepts related to describing a distribution of numbers:

  • Creating histograms to show score distribution.

  • Describing the shape of the distribution (unimodal, bimodal, multimodal, symmetrical, skewed).

  • Measuring central tendency (mean, median, mode).

  • Describing variability (variance or standard deviation).

• These methods describe the entire sample, whereas Z-scores describe individual scores.

Individual Scores and Standard Deviations

• The method to describe individual scores involves standard deviations.

• The concept relates to the earlier discussion on outliers.

Outliers Revisited

• The process for determining outliers involves finding the standard deviation and determining a cutoff point for 2.5 standard deviations from the mean.

• Example: mean = 50, standard deviation = 10.

  • 10 * 2.5 = 25

  • 50 - 25 = 25

  • 50 + 25 = 75

  • Anything lower than 25 or higher than 75 is an outlier.

• 25 is 2.5 standard deviations below the mean, and 75 is 2.5 standard deviations above the mean.

Calculating Standard Deviations from the Mean

• To calculate how many standard deviations a number is from the mean, follow these steps:

  1. Subtract the mean of the sample from the raw score.

  2. Divide the result by the standard deviation.

• Example: Given a distribution with a mean of 50 and a standard deviation of 10, calculate how many standard deviations away from the mean is 25.

  • 25 - 50 = -25

  • -25 / 10 = -2.5

  • 25 is 2.5 standard deviations below the mean.

• The negative sign indicates the number is below the mean.

Examples of Calculating Standard Deviations

• Calculating for 75 in the same distribution (mean of 50 and a SD of 10):

  • 75 - 50 = 25

  • 25 / 10 = 2.5

  • 75 is 2.5 standard deviations above the mean.

• Calculating for 70 in the same distribution:

  • 70 - 50 = 20

  • 20 / 10 = 2

  • 70 is 2 standard deviations above the mean.

• This means that 70 is twice the average amount above the mean.

Z-Scores Explained

• A Z-score represents how many standard deviations above or below the mean a score is located.

Applying Z-Scores to the Initial Question

• Revisiting the initial question with additional information:

  • Peter scored 95 in professor 1’s course, where the mean was 85 and the standard deviation was 5.6.

  • Mary scored 90 in professor 2’s course, where the mean was 80 and the standard deviation was 6.2.

• To determine who has bragging rights, convert both scores to Z-scores.

Calculating Peter's Z-Score

• Peter’s data: score = 95, mean = 85, standard deviation = 5.6.

  1. Subtract mean from the raw score: 95 - 85 = 10

  2. Divide by the standard deviation: 10 / 5.6 = 1.79

• Peter’s Z-score is 1.79, meaning he is 1.79 standard deviations above the mean in his distribution.

Calculating Mary's Z-Score

• Mary’s data: score = 90, mean = 80, standard deviation = 6.2.

  1. Subtract mean from the raw score: 90 - 80 = 10

  2. Divide by the standard deviation: 10 / 6.2 = 1.61

• Mary’s Z-score is 1.61, meaning she is 1.61 standard deviations above the mean in her distribution.

Comparing Z-Scores

• Peter has bragging rights because his Z-score (1.79) is higher than Mary’s (1.61).

• Peter’s score was more “extreme” than Mary’s score.

• Converting to Z-scores is called “standardizing scores” because it transforms scores in a way that allows comparison across different distributions.

Key Facts about Z-Scores

• The absolute value of a Z-score determines how extreme it is, regardless of the sign.

  • For example, -5.2 is more extreme than +2.5.

• The sign indicates whether the score is below or above the mean.

Interpreting Z-Scores

• A Z-score is positive if the raw score is above the mean and negative if it is below the mean.

• The larger the absolute value of the Z-score, the more unusual the raw score.

• Example:

  • A Z-score of +4 indicates the raw score is unusually far above the mean.

  • A Z-score of -4 indicates the raw score is unusually far below the mean.

  • Both scores are equally unusual but in different directions.

Z-Scores and Standard Deviations

• Z-scores are the same as standard deviations.

• A raw score with a Z-score of 0 is 0 standard deviations from the mean, meaning it is the mean.

• Raw scores with Z-scores more extreme than 2.5 (< -2.5 or > +2.5) are considered outliers.

Example with Variance

• Given a set of data with mean (M) = 100 and variance (s^2) = 25, convert scores of 75, 90, and 105 to Z-scores.

• First, convert variance to standard deviation:

  • s = \sqrt{25} = 5

More Examples with Z-Scores

• Using M = 100 and s = 5

  • For 105:

    • 105 - 100 = 5

    • 5 / 5 = 1

    • Z-score = 1 (1 standard deviation above the mean).

  • For 90:

    • 90 - 100 = -10

    • -10 / 5 = -2

    • Z-score = -2 (2 standard deviations below the mean).

  • For 75:

    • 75 - 100 = -25

    • -25 / 5 = -5

    • Z-score = -5 (5 standard deviations below the mean).

Additional Examples

• Given M = 120 and s^2 = 49, thus s = 7

  • For 130:

    • 130 - 120 = 10

    • 10 / 7 = 1.43

    • Z-score = 1.43 (1.43 standard deviations above the mean).

  • For 95:

    • 95 - 120 = -25

    • -25 / 7 = -3.57

    • Z-score = -3.57 (3.57 standard deviations below the mean).

  • For 121:

    • 121 - 120 = 1

    • 1 / 7 = 0.14

    • Z-score = 0.14 (0.14 standard deviations above the mean).

Normal Distributions and Z-Scores

• The lecture mentions the importance of having mostly “normal” distributions.

• The next module will cover further applications of Z-scores in the context of normal distributions.