Central Tendency and Variability in Height Data: Mean, Median, Mode, Range, Variance, and Standard Deviation

Data Overview and Setup

• Variable: height
• Frequency table observed: heights 60, 61, 62, 63, 64, 65 with frequencies 2, 3, 5, 5, 3, 2 respectively
• Total observations: N = 20
  • Verification: sum of frequencies = 2 + 3 + 5 + 5 + 3 + 2 = 20
• Histogram interpretation: height of bar corresponds to how many people have that height; translates the frequency table into a distribution of heights

Central Tendency: Mean, Median, and Mode

• Mean (mathematical average)
  • Formula: $\bar{x} = \frac{\sum f<em>i x</em>i}{n}$
  • Computation from the data:
  • Numerator: $60\times 2 + 61\times 3 + 62\times 5 + 63\times 5 + 64\times 3 + 65\times 2 = 1250$
  • Denominator: $n = 20$
  • Result: $\bar{x} = \frac{1250}{20} = 62.5$
• Median
  • For an even number of observations, the median is the average of the two middle values
  • With N = 20, the two middle positions are the 10th and 11th values
  • From the ordered data, the middle values are 62 and 63
  • Hence: $\text{Median} = \frac{62 + 63}{2} = 62.5$
• Mode
  • Definition: most frequent value(s)
  • In this dataset, 62 and 63 each occur 5 times → bimodal with modes at 62 and 63
• Relationship to symmetry
  • A perfectly symmetric distribution has mean = median
  • If unimodal and symmetric, the mode often coincides with the center; if bimodal symmetric, there can be two modes
  • In the example, mean = median = 62.5; mode can be one or two values depending on the peak structure

Influence of Symmetry and Outliers

• Two distributions with the same central tendency but different spreads illustrate key ideas:
  • If the distribution is symmetric, the mean and median are equal
  • Outliers affect the mean more than the median; skewness shifts the mean away from the center
  • If outliers balance (roughly symmetric but with heavy tails), the mean and median can still be close
• Symmetry and the mode
  • In a symmetric unimodal distribution, the mode is at the center (single mode)
  • In a symmetric but bimodal distribution, there can be two modes

Variability: Range, Variance, and Standard Deviation

• Importance of variability
  • Central tendency alone (mean/median/mode) may hide how spread out the data are
  • Two distributions can have the same center but different spreads
• Range
  • Definition: $\text{Range} = \max x<em>i - \min x</em>i$
  • Example: If heights range from 60 to 65, $\text{Range} = 65 - 60 = 5$
  • Alternative description: “from 60 to 65”
  • Note: Range ignores how data are distributed between min and max
• Deviations and the sum of squares
  • Deviation from the mean for each observation: $x_i - \bar{x}$
  • Distinguish sign: below mean gives negative values, above mean gives positive values
  • To avoid cancellation, square deviations: $(x_i - \bar{x})^2$
  • Sum of squared deviations (Sum of Squares): $\sum<em>{i=1}^n (x</em>i - \bar{x})^2$
• Sample variance and standard deviation
  • Sample variance (with Bessel's correction): $s^2 = \frac{\sum<em>{i=1}^n (x</em>i - \bar{x})^2}{n-1}$
  • In the worked example, the sum of squared deviations is 64 and n = 5, so: $s^2 = \frac{64}{5-1} = \frac{64}{4} = 16$
  • Sample standard deviation: $s = \sqrt{s^2} = \sqrt{16} = 4$
• Quick intuition
  • The standard deviation measures the typical distance of a score from the mean
  • Smaller SD = data are more tightly clustered around the mean; larger SD = more spread
• Population parameters (for reference)
  • Population variance: $\sigma^2 = \frac{\sum (x_i - \mu)^2}{n}$
  • Population standard deviation: $\sigma = \sqrt{\sigma^2}$

Worked Example: Step-by-Step from Frequency Table to Summary Stats

• Data setup recap: heights with frequencies 60(2), 61(3), 62(5), 63(5), 64(3), 65(2) → N = 20
• Mean recap: $\bar{x} = \frac{60\cdot 2 + 61\cdot 3 + 62\cdot 5 + 63\cdot 5 + 64\cdot 3 + 65\cdot 2}{20} = \frac{1250}{20} = 62.5$
• Median recap: for N = 20, middle values are 62 and 63 → $\text{Median} = \frac{62 + 63}{2} = 62.5$
• Mode recap: 62 and 63 both occur 5 times → two modes (62, 63)
• Range recap: max = 65, min = 60 → $\text{Range} = 5$
• Variability example (sum of squared deviations)
  • Suppose $\sum<em>{i=1}^n (x</em>i - \bar{x})^2 = 64$ for this illustrative subset of data
  • Then: $s^2 = \frac{64}{5-1} = 16, \quad s = \sqrt{16} = 4$
• Interpretations
  • Mean = 62.5, Median = 62.5; Mode depends on peak structure (here could be two modes)
  • Range = 5; SD = 4 in the worked numerical example

Quick Formulas Recap

• Central tendency
  • $\bar{x} = \frac{\sum f<em>i x</em>i}{n}$
  • Median: middle value(s) logic for even/odd n
  • Mode: most frequent value(s)
• Variability
  • Range: $\text{Range} = \max x<em>i - \min x</em>i$
  • Variance (sample): $s^2 = \frac{\sum<em>{i=1}^n (x</em>i - \bar{x})^2}{n-1}$
  • Standard deviation (sample): $s = \sqrt{s^2}$
  • Population variance: $\sigma^2 = \frac{\sum (x_i - \mu)^2}{n},\quad \sigma = \sqrt{\sigma^2}$
• Key takeaway
  • Symmetric distributions tend to have mean ≈ median; unimodal symmetric distributions often have mode at the center; symmetry does not guarantee a single mode
  • The two core pieces to describe a distribution well are the central tendency (mean/median/mode) and the variability (range, variance, standard deviation)