Normal Distributions

Normal distributions

  • Notation: N(μ,σ2)N(\mu, \sigma^2)
  • Center: μ\mu; Mean=Median=Mode=μ\text{Mean}=\text{Median}=\text{Mode}=\mu
  • Shape: Bell-shaped; symmetric; tails extend to infinity
  • Spread: determined by σ\sigma; variance σ2\sigma^2

WHY DOES THIS MATTER?

  • Distributions show how data look
  • Statistical tests assume distributions; commonly rely on normality

CHARACTERISTICS OF A NORMAL DISTRIBUTION

  • Symmetrical; sides mirror each other
  • Tails never touch the x-axis; extend to infinity
  • Mean=Median=Mode=μ\text{Mean}=\text{Median}=\text{Mode}=\mu
  • Bell-shaped; shape depends on variability
  • Spread reflects sample variation via σ\sigma (variance σ2\sigma^2)

SHAPES (KURTOSIS)

  • Kurtosis measures peakedness/flatness
  • Leptokurtic: highly peaked
  • Mesokurtic: intermediate (normal-ish)
  • Platykurtic: flat

SKEWED DISTRIBUTIONS

  • Not symmetric
  • Positive skew: tail to the right
  • Negative skew: tail to the left

ASSESSING SKEWNESS

  • High skew: skew<−1\text{skew} < -1 or skew>1\text{skew} > 1
  • Moderate skew: −1<skew<−0.5-1 < \text{skew} < -0.5 or 0.5<skew<10.5 < \text{skew} < 1
  • Approximately symmetric: −0.5≤skew≤0.5-0.5 \le \text{skew} \le 0.5

BIMODAL DISTRIBUTIONS

  • Two modes; report both
  • Reporting a mean is unlikely to be appropriate

SUMMARY

  • Recognise a normal distribution and its role in statistics
  • Differentiate normal curve shapes
  • Differentiate positive vs negative skew
  • Assess skewness
  • Recognise a bimodal distribution