Notes on Central Tendency, Distribution Shape, and Variation

Measures of Central Tendency

• Central tendency: measures that describe the middle of a dataset.

• Major measures: mean (average), median, and mode. All start with 'm' in naming, but they have different meanings and uses.

• Population vs. sample distinction matters for notation and interpretation, not for the basic idea of the mean.

Mean

• Also called the average in everyday language.

• Notation:

  • Population mean: $\mu$

  • Sample mean: $\bar{x}$

• Formulas:

  • Population mean: $\mu = \frac{1}{N} \sum_{i=1}^{N} x_i$

  • Sample mean: $\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i$

• Summary of notation:

  • When dealing with populations, we commonly use Greek symbols (e.g., $\mu$, $\sigma$).

  • When dealing with samples, we use Latin letters (e.g., $\bar{x}$, $s$).

• Why two symbols?

  • The formulas look the same, but the interpretation differs: a population mean is a fixed (but unknown) parameter; a sample mean varies from sample to sample.

  • If you take different samples from the same population, you’d expect different $\bar{x}$ values; the population mean $\mu$ would be the same (if you had the whole population, there’d be no sampling variability).

• Example: Ages in an entire family (population).

  • Suppose the dataset yields a population mean of about $\mu \approx 41.86\,$ (rounded to 42 in context).

  • Interpretation: the average age of the family is around 42 years.

  • Note on interpretation: the mean describes the center, but it may not capture the dataset well if there are outliers or a skewed distribution.

• Example: Sample mean context (distance to campus for a sample of faculty).

  • A sample mean might be something like $\bar{x} = 12.2\,\text{units}$ (e.g., minutes or miles, depending on data).

  • The suitability of the mean as a descriptor depends on the dataset; sometimes the sample mean better describes the dataset than the population mean in practice.

• Quick takeaways:

  • The mean uses every value in the dataset, which is an advantage (no data are ignored).

  • The mean is sensitive to outliers: extreme values can pull the mean toward them.

Median

• Definition: the middle value of a dataset when ordered from smallest to largest.

• How to find it:

  • If the number of data points n is odd, the median is the middle value.

  • If n is even, the median is the average of the two middle values.

• Formula for the two-middle case (when in order):

  $\text{median} = \frac{x_{(\frac{n}{2})} + x_{(\frac{n}{2}+1)}}{2}\; (n\text{ even})$

• For odd n: median is the value at position ($x_{((\frac{n+1}{2}))}$).

• Terminology:

  • The median is sometimes called the midpoint of the data. To avoid confusion with the mean, the term midpoint is often used when two middle values are averaged.

• Example concepts:

  • For a small dataset with 5 numbers, the median is the third value after sorting.

  • For an even-sized dataset (e.g., 6 numbers), the median is the average of the 3rd and 4th values, which may yield a value like $11.5$ (not always ending in .5, but commonly when the central values are consecutive).

• Why use the median?

  • The median can better describe the center when data are skewed or contain outliers, since it is not pulled toward extreme values.

Mode

• Definition: the value that occurs most often in the dataset.

• Key points:

  • If no value repeats, there is no mode.

  • A dataset can have more than one mode:

  • Two modes: bimodal

  • Three modes: trimodal

  • More than two: multimodal (sometimes used informally; technically “multimodal”)

• Practical note:

  • The mode indicates the most frequent value but is not always informative about the dataset’s overall shape or center.

• Example:

  • A dataset where 14 occurs most often has a mode of 14.

Relationship of central tendency measures to data shape

• In symmetric distributions (bell-shaped), the mean, median, and mode are all equal (or very close): $\mu \approx \text{median} \approx \text{mode}$

• In uniform distributions, the mean and median are equal, but the mode may be undefined or may occur at multiple values.

• In skewed distributions:

  • Skewed right (outliers to the right) typically has \mu > \text{median}

  • Skewed left (outliers to the left) typically has \mu < \text{median}

Measures of Variation and Distribution Shape

Range

• Definition: the difference between the maximum and minimum values in the dataset.

• Formula:
  $\text{Range} = \max_i(x_i) - \min_i(x_i)$

• Example interpretation:

  • If Company A salaries have range $=10$ (thousand dollars) and Company B salaries have range $=35$, Company B has a wider spread in salaries, suggesting greater variability.

Deviation, Variance, and Standard Deviation

• Deviation:

  • For a given data value, the deviation from the mean is:
    $\quad d_i = x_i - \mu\quad\text{(population)}$
    or
    $\quad d_i = x_i - \bar{x}\quad\text{(sample)}$

  • Sign indicates whether the value is below (negative) or above (positive) the mean.

• Population variance and standard deviation:

  • Variance (population):
    $\sigma^2 = \frac{1}{N}\sum_{i=1}^{N} (x_i - \mu)^2$

  • Standard deviation (population):
    $\sigma = \sqrt{\sigma^2}$

• Why square deviations?

  • To avoid cancellation of positive and negative deviations; variance measures average squared distance from the mean.

• Sample variance and standard deviation (with Bessel's correction):

  • Sample variance:
    $s^2 = \frac{1}{n-1}\sum_{i=1}^{n} (x_i - \bar{x})^2$

  • Sample standard deviation:
    $s = \sqrt{s^2}$

• Bessel's correction: divide by (n-1) (not by (n)) to account for the fact that a sample typically underestimates the population variability; provides a less biased estimate of the true variability.

• Note: The correction is specific to samples; for populations you divide by (N).

• Intuition: with a smaller bottom (n-1), the estimate of spread is a bit larger to allow for sampling variability (and to reflect the fact that the sample mean is used instead of the true population mean).

• Practical takeaway:

  • Standard deviation tells you, on average, how far data values are from the mean.

  • Smaller standard deviation means data are tightly clustered around the mean; larger standard deviation means more spread.

• Worked example (population):

  • Data (population) of salaries A: mean $\bar{x} = 41.5\,(\text{thousand})$ (example value).

  • Deviations: subtract the mean, square, and sum; suppose the sum of squared deviations equals 7,? (example step shown in the lecture).

  • Variance: $\sigma^2 = \frac{1}{N}\sum (x_i - \mu)^2$ (In the example, this yielded a value of $88.85$)

  • Standard deviation: $\sigma = \sqrt{\sigma^2}$ (In the example, this was approximately $9.43$)

  • Interpretation: average deviation in the chosen units.

• Worked example (sample, with steps):

  • Data (sample) from eight players recovering from a concussion.

  • Mean: $\bar{x} = 39.5$

  • Deviations: subtract 39.5, square, and sum.

  • Sum of squared deviations (example): 56.0 (illustrative).

  • With Bessel's correction (n = 8, n-1 = 7):

  • Sample variance: $s^2 = \frac{\text{sum of squared deviations}}{n-1}$ (For example, if sum of squared deviations is $56.0$ and $n=8$, then $s^2 = 8.0$)

  • Sample standard deviation: $s = \sqrt{s^2}$ (In the example, this was approximately $2.83\,\text{days}$)

  • Interpretation: On average, recovery times vary by about $2.83\,\text{days}$ around the mean of $39.5\,\text{days}$.

• Important distinction (contextual):

  • Population measures use the entire group; sample measures use a subset and adjust with Bessel's correction to better estimate the population value.

• Practical example from the lecture:

  • Concussion data: mean 39.5, standard deviation $\approx 13.3$ (days).

  • Interpretation: On average, recovery times vary by about $13.3\,\text{days}$ from the mean; a larger spread indicates more individual variation in recovery times.

The Empirical Rule (for approximately symmetric data)

• If the data are approximately symmetric (bell-shaped):

  • About 68% of data lie within one standard deviation of the mean: $P(|X - \mu| \le \sigma) \approx 0.68$

  • About 95% lie within two standard deviations: $P(|X - \mu| \le 2\sigma) \approx 0.95$

  • About 99.7% lie within three standard deviations: $P(|X - \mu| \le 3\sigma) \approx 0.997$

• What this means in practice:

  • If a distribution is symmetric, you can estimate how much data falls in these bands without computing every value.

  • If data lie outside two standard deviations from the mean in a symmetric shape, they are in the outer 5% (extreme values).

• Important caveat:

  • The empirical rule applies to symmetric data, not to skewed data; for skewed data, the percentages will not match these exact values.

• Conceptual analogy used in class:

  • Skittle example: with 100 items and 5 poisoned, the event of drawing a poisoned item is extremely rare (outside the 5% tail); this illustrates how we think about tails and probability bounds in symmetric contexts.

Practical Notes and Strategies

• Always consider all three measures (mean, median, mode) to understand data; they can tell different stories depending on shape and outliers.

• Outliers affect the mean more than the median; this is a key reason to report multiple measures.

• Range can be informative but is sensitive to extreme values; it does not capture how values are distributed between min and max.

• Use weighted means when different observations contribute unequally (e.g., GPA weighted by credit hours):

  • Weighted mean formula:
    $\bar{x}_w = \frac{\sum_i w_i x_i}{\sum_i w_i}$

  • Example (GPA): if course grades are weighted by credit hours, the total weighted sum divided by total credits gives the GPA; a worked example in class yielded approximately $3.29$ (which rounds to about 3.3) for a GPA.

• The choice of descriptor depends on the data: for skewed data or data with outliers, the median or mode may describe the center better than the mean; for relatively clean, symmetric data, the mean is often informative.

• A note on practice:

  • Formulas are important concepts to understand; you’ll perform by-hand calculations a few times to build intuition, then use software (e.g., Excel) for larger datasets.

• Summary of notations:

  • Population: mean $\mu$, standard deviation $\sigma$, variance $\sigma^2$ (and sometimes a population-specific formula).

  • Sample: mean $\bar{x}$, standard deviation $s$, variance $s^2$, with the key correction (divide by (n-1)) known as Bessel's correction.

• Quick interpretive guideline:

  • If mean is much smaller than median, there may be a left (low-end) outlier pulling the mean down.

  • If mean is much larger than median, there may be a right (high-end) outlier pulling the mean up.

Appendix: Notation Quick Reference

• Notation used for populations: $\mu, \sigma, \sigma^2$, etc.

• Notation used for samples: $\bar{x}, s, s^2$, etc.

• X-bar ($\bar{x}$) vs Mu ($\mu$): sample vs population means, respectively.

• The summation symbol $\sum$ is used to denote adding a series of numbers.

• Order statistics: $x_{(k)}$ denotes the k-th smallest value in the ordered data.

• Deviation notations: $d_i = x_i - \mu\quad\text{(population)}$ or $$d_i = x_i -