Psychological Statistics Review

Psych Stats: Standardized Scores

Introduction to Standardized Scores

• A standardized score is expressed in terms of the number of standard deviations from the mean.

• Standardized scores indicate how many standard deviations a particular score is above or below the mean.

Calculating Standardized Scores (Z-scores)

• The formula for calculating a standardized score (Z-score) is given as:

  • $Z = \frac{x - m}{S}$

    • where:

    • $x$ = individual score

    • $m$ = mean of the distribution

    • $S$ = standard deviation of the distribution

Example Calculations

• Football example:

  • Given: 2 sacks; mean ($m$) = 5; standard deviation ($S$) = 1

  • Calculation:

  • $Z = \frac{2 - 5}{1} = -3$

• Cross Country example:

  • Given: Time of 17.5; mean = 19; standard deviation = 2

  • Calculation:

  • $Z = \frac{17.5 - 19}{2} = -0.75$

Details on Standardized Scores

• For the example of Blue (dog):

  • Age = 29.5 years; mean ($m$) = 13.41 years; SD = 2.36 years

  • Calculation:

  • $Z = \frac{29.5 - 13.41}{2.36} = 6.81779661$

• For Jake (dog):

  • Age = 12.75 years; SD = 2.36 years; mean = 13.41 years

  • Calculation:

  • $Z = \frac{12.75 - 13.41}{2.36} = -0.2796$

Additional Standard Score Configurations

• When converting raw scores to a new distribution, the new mean ($M{new}$) and new standard deviation ($S{new}$) should be set before conversion:

  • Standard answer: 1. Convert raw numbers to Z-scores using:

  • $X<em>{new} = Z \cdot S</em>{new} + M_{new}$

• Example conversion using MIQRAW 250:

  • $X_{raw} = 265$; new mean = 100; new SD = 15

  • Step 1: Convert:

  • $Z = \frac{265 - 250}{50} = 0.3$

  • Step 2: New distribution:

  • $X_{new} = 3 \cdot 15 + 100 = 104.5$

Review of Z-scores

• The standard score represents how far, on average, a score is from the mean.

• Z-scores allow us to create a standard distribution where:

  • The mean of this distribution is always 0.

  • The standard deviation is always 1.

• Standardized scores enable comparison across different datasets by establishing relative positions.

Probability Concepts

• Probability: Formal definition relates to the proportion of all possible outcomes that correspond to a certain event.

• Notation:

  • $P(x)$ defines the probability of obtaining outcome $x$.

Example Probability Calculations

• Probability of drawing a spade:

  • $P(spade) = \frac{13}{52} = 0.25$

• Similar calculations for coins and marbles:

  • Probability of heads from a coin: $P(H) = \frac{1}{2} = 0.5$

  • Probability of marble color example:

  • Given: 3 blue, 2 red, 7 black

  • Probabilities:

    • $P(blue) = \frac{3}{12} = 0.25$

    • $P(red) = \frac{2}{12} = 0.1667$

    • $P(black) = \frac{7}{12} = 0.5833$

Normal Distribution

• Normal Distribution: Characterized by a bell curve where:

  • It is symmetrical about the mean with mean, median, and mode all equal.

  • The relative frequency is predictable based on mean ($m$) and standard deviation ($S$).

Key Attributes of Normal Distribution

• The total area under the curve equals 1.

• The distribution's shape is determined by mean and standard deviation.

• Properties of standard normal distribution:

  • Approximately 68.26% of all scores fall within +/- 1 standard deviation from the mean.

Applying Normal Distribution Concepts

• Example:

  • Given a normal distribution with mean ($M$) = 35 and standard deviation ($S$) = 7, calculate:

  • P(X < 45)

  • First find Z-score:

  • $Z = \frac{45 - 35}{7} = 1.43$

  • Look up area for Z = 1.43.

Hypothesis Testing Basics

• Defines a method to determine if a pattern or effect in data is real, typically involves:

  1. Stating a null hypothesis (Ho): all conditions are equal.

  2. Alternative hypothesis (H₁): rejecting Ho, indicating conditions differ.

Errors in Hypothesis Testing

• Type I Error: Rejecting Ho when it is true (False Positive).

• Type II Error: Failing to reject Ho when Ho is false (False Negative).

Guidelines for Conducting Hypothesis Testing

1. Determine the statistical test to employ.

2. State hypotheses.

3. Set rejection criteria.

4. Collect and analyze the data.

5. Compute test statistics and interpret results.

Z-tests

• Utilize Z-scores as test statistics to evaluate if a sample is from a known population.

  • Mutually exclusive hypotheses apply:

  • Ho: the sample is from a known population.

  • H1: the sample is not from a known population.

Sampling Distributions

• A vital foundation for statistical inference tests, as it helps estimate probabilities for sample means when sampling from a population with replacement.

• Complete understanding of sampling distributions supports hypothesis testing and further analytical methods in statistics.