Standard Deviation, Z-Scores, and Percentiles

Mean and deviations

• The mean (average) is a central value around which other numbers deviate.
• Example: If the mean is 50 and a data point is 60, the deviation is +10 from the mean:
  $d<em>i = x</em>i - \bar{x}$ with (xi = 60) and (\bar{x} = 50), so $d</em>i = 60 - 50 = 10.$
• Deviations can be positive or negative depending on whether the value is above or below the mean.
• To talk about the size of the deviation (the magnitude), we use the absolute value: $|d<em>i| = |x</em>i - \bar{x}|.$
• When asked to estimate a typical deviation, you might encounter choices like 1, 6, or 20; in the example, 6 is chosen as a reasonable estimate because it’s in the ballpark of the observed deviations, while 1 is too small and 20 is too large.
• For a pair of histograms (top vs bottom), you might be asked to pick a representative deviation size that seems typical from the visual spread; this is a rough, eyeball estimate rather than a precise calculation.

What standard deviation is and why we do it

• The standard deviation is a single number that summarizes how spread out the data are about the mean.
• The core idea is to quantify the typical deviation from the mean in a way that is:
  • Sensitive to large deviations (larger deviations contribute more).
  • All deviations are treated in a consistent, non-canceling manner (hence squaring).
• The steps to move from deviations to a standard deviation involve squaring deviations, summing, and then averaging (with a correction for bias in small samples):
  • Deviations: $d<em>i = x</em>i - \bar{x}$
  • Sum of deviations (should be zero): $\sum<em>{i=1}^n d</em>i = 0.$
  • Square the deviations: $d<em>i^2 = (x</em>i - \bar{x})^2.$
  • Sum of squares (SS): $SS = \sum<em>{i=1}^n (x</em>i - \bar{x})^2.$
  • Average the squared deviations and take the square root to get the standard deviation. For a sample, the divisor is (n-1):
  • Sample variance: $s^2 = \frac{SS}{n-1},$
  • Sample standard deviation: $s = \sqrt{\frac{SS}{n-1}}.$
• Why divide by (n-1) instead of (n)? Using (n-1) gives an unbiased estimator of the population variance when you are working from a sample. (Division by (n) would systematically underestimate the true variance when estimating from samples.)
• If you were working with the entire population, you would divide by (n) to get the population standard deviation: $\sigma = \sqrt{\frac{SS}{n}}.$

Step-by-step: how to compute the standard deviation (sample)

• Step 1: Compute the mean (\bar{x}) of the data set.
• Step 2: For each data point (xi), compute the deviation $d</em>i = x_i - \bar{x}.$
• Step 3: Check that the deviations sum to zero (as a consistency check): $\sum<em>{i=1}^n d</em>i = 0.$
• Step 4: Square each deviation: $d<em>i^2 = (x</em>i - \bar{x})^2.$
• Step 5: Sum the squared deviations: $SS = \sum<em>{i=1}^n (x</em>i - \bar{x})^2.$
• Step 6: Compute the sample variance: $s^2 = \frac{SS}{n-1}.$
• Step 7: Compute the sample standard deviation: $s = \sqrt{\frac{SS}{n-1}}.$
• Note: In the example, an intermediate value like 36 may appear in discussion; the key is following the above steps and using (n-1) in the denominator for the sample standard deviation.

Z-scores: standardizing values

• Z-score measures how many standard deviations an observation is from the mean.
• Population version (uses population mean and population std. dev.):
  • $z = \frac{x - \mu}{\sigma}.$
• Sample version (uses sample mean and sample std. dev.):
  • $z = \frac{x - \bar{x}}{s}.$
• Converting back from a z-score to an original value:
  • Population: $x = \mu + z\sigma.$
  • Sample: $x = \bar{x} + zs.$
• Interpretation: A larger |z| means the value is farther from the mean in units of standard deviations.
• Example setup (apple): If the mean is (\mu) and the standard deviation is (\sigma), the z-score for an apple weighing 135 g is
  • $z = \frac{135 - \mu}{\sigma}.$
• Practical use: You can do these conversions with or without the formula sheet, and you can interpret the number of standard deviations away from the mean for decision making or comparison.

The 95% rule (empirical rule) for z-scores

• About 95% of data fall within two standard deviations of the mean:
  • $P(|Z| \le 2) \approx 0.95.$
• This means if you compute z-scores for data, roughly 95% should satisfy (-2 \le z \le 2).
• More generally:
  • A z-score of 3 corresponds to three standard deviations above the mean: you are at the location z = 3 when the standardized value is three.
• The empirical rule helps in quick assessments of spread and outliers.

Conversions between x-values and z-scores (summary)

• To go from x to z: use $z = \frac{x - \bar{x}}{s}\quad\text{(sample)}\text{ or }\quad z = \frac{x - \mu}{\sigma}\quad\text{(population)}.$
• To go from z to x: use $x = \bar{x} + zs\quad\text{(sample)}\text{ or }\quad x = \mu + z\sigma\quad\text{(population)}.$
• These conversions allow you to compare values measured on different scales and to locate where a data point sits relative to the mean.

Percentiles: locating positions in the distribution

• A percentile describes the value below which a given percentage of data falls.
• Formal idea: the p-th percentile (x_p) satisfies
  • $P(X \le x_p) = p.$
  • For example, the 15th percentile is the value such that 15% of the data are at or below that value; the 70th percentile is where 70% are at or below.
• Practical usage: to find a percentile from data, sort the data and identify the value at the position corresponding to that percentile (or interpolate between values if needed).
• The transcript indicates determining specific percentile values on the x-axis by starting from the left and accumulating data to reach the desired percentile, which is the intuitive approach when reading from a histogram or distribution.

Quick recap and how these ideas connect

• Deviations quantify how far each data point is from the center; the collection of these deviations motivates the standard deviation as a single-number summary of spread.
• Squaring deviations eliminates cancellation and emphasizes larger deviations, leading to the sum of squares (SS) and the variance.
• The difference between dividing by (n-1) (sample) vs. (n) (population) reflects bias correction when estimating population variance from a sample.
• Z-scores standardize data to units of standard deviations, enabling comparisons across different datasets and linking to the normal-distribution intuition (empirical rule).
• Percentiles provide a way to describe position within the distribution beyond the mean and standard deviation, useful for ranking and threshold decisions.

Practice prompts inspired by the transcript

• If the mean is 50 and a data point is 60, what is the deviation and its absolute value?
• Given a small sample, why is the denominator in the variance formula often (n-1) rather than (n)?
• Compute the z-score of a value given a mean and standard deviation, and interpret its magnitude.
• State the 95% rule in terms of z-scores and in terms of percentages of data.
• Describe how to locate the 15th and 70th percentiles on a distribution or histogram.