Week 4 - Standard Scores

Recap: Describing Data

1. Shape of a distribution: number of modes, skew, tail

2. Central Tendency: What is typical? Give me a single number summary.

3. Variability: How spread out are the scores?

• How well does that single number represent all scores?

  • IQR for the range around Median (Box plots show this)

  • SS (sum of squares), Variance, SD for the variability around Mean

• Use visualization and descriptive stats together to get a better understanding of the data

For a unimodal, symmetric distribution, which of the following is true about its central tendency?

Mean = Median = Mode

Which descriptive statistics are most appropriate for describing a normal distribution?

Mean and Standard Deviation

Normal Distribution: Mean and SD

• When you know M and SD, you can recreate the normal distribution

• Mean tells us about the center (location)

• SD tells about the variability around the mean (spread)

Negative Skew, unimodal distribution

Mean < Median < Mode

Positive Skew, unimodal distribution

Mode < Median < Mean

A Normal Distribution Maps SDs to Probability

• Knowing the Mean and SD lets you reconstruct the normal distribution

• Range of SDs from the mean maps to a probability (Area under the curve)

  • Range of SDs from the mean → Probability

    • find probability for any range of SDs

  • Probability → Range of SDs from the mean

    • find cutoff SDs for any probability

The 69-95-99.7 Rule

• M ± 1SD: ~68%

• M ± 2SD: ~95%

• M ± 3SD: ~99.7

Apple weights follow a normal distribution with a mean of 150g and a standard deviation of 15g. What percentages of apples weigh between 135g and 165g?

About 68%

Apple weights follow a normal distribution with a mean of 150g and a standard deviation of 15g. About 95% of apple are expected to weigh:

120g to 180g

Scores and Probabilities (Pt 1)

• Range of scores maps to a probability (area under the curve)

  • Cutoff scores → (cutoff SDs from the mean) → Probability

    • find probability for any range of scores (ex. probability of heights between 5ft to 6ft)

  • Probability → (cutoffs SDs from the mean) → cutoff scores

    • find cutoff scores for any probability (ex. the height marking the top 10%)

Standard Scores (z Scores): SDs from the mean

• expressed as the relative standing from the mean, in units of typical deviation (SD)

• enable comparing scores from different distributions

• measures how many standard deviations a specific data point is away from the mean (average) of a dataset

Your standardized exam score (z) is 3. You performed:

better than most students (three standard deviations away from the mean)

An apple’s weight has a z score of -2. This apple is:

lighter than most apples

An apple’s weight has a z score of -2. What percentage of apples are lighter than this apple?

2.5%

Standard Scores (z scores) from Raw Scores

• z scores convert raw scores into:

  • relative standing from the mean

  • in units of standard deviation

• standard score (z) = (raw score - mean) / (standard deviation)

  • Step 1: Centering: subtract the mean → relative standing from the mean

  • Step 2: Scaling: divide by the SD → expressed in SD units

Step 1. Centering

• An apple weighs 162 g (M = 150, SD = 15 g)

• raw score - mean

• = 162 - 150 = 12

• This apple weighs 12 g more than the average

  • → distance from the mean = 12

Step 2. Scaling

• An apple weighs 162 g (M = 150, SD = 15 g)

• (raw score - mean) / standard deviation

• = (162 - 150) / 15 = 12 / 15 = 4 / 5 = 0.8

• This apple weighs above average. but by less than a typical deviation (SD)

Apples to Oranges: Comparing Different Scores

• By standardizing scores, you can compare scores from different distributions

  • Household incomes in different countries

  • Test scores from different classes

  • Cognitive performance in different age groups

Comparing Scores: Example Pt 1

• Which test did Zoey do better in?

  • SAT = 1350 (M = 1051, SD = 211)

  • ACT = 30 (M = 21.00, SD = 5.20)

→ ACT

SAT: (1350 - 1051) / 211 =1.417 standard deviations away from the mean

ACT: (30 - 21) / 5.20 =1.731 standard deviations away from the mean

On calculator: use normalcdf function

• Percentage below a z-score → normalcdf(-9999, z, 0, 1)

• Percentage above a z-score → normalcdf(z, 9999, 0, 1)