Lecture 03 Univariate Data Exploration

Quantitative Data: Approach 2

• Location: Mean
• Spread: Standard Deviation
  • Roughly the average distance values fall from the mean.

Example

Numbers	Mean	Standard Deviation
100, 100, 100, 100, 100	100	0
90, 90, 100, 110, 110	100	10

• Both sets have the same mean of 100.
• Set 1: No variability (all values equal the mean).
• Set 2: Variability exists (four values are 10 points away from the mean).

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Density Curves

• Example: Histogram of vocabulary scores for 947 seventh graders.
• The smooth curve over the histogram is a mathematical model for the distribution.

Key Properties

• Always on or above the horizontal axis.
• Total area under the curve is exactly 1.
• Area under the curve represents the proportion of observations in a range.
• Median: Equal-areas point.
• Mean: Balance point.

Example: Shaded Area

• Shaded area represents scores ≤ 6.0.
• Proportion = 0.293 (adjusted for density curve).

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Relation of Shape and Central Location

• Symmetric: Balanced around the mean.
• Skewed Right: Tail extends to the right.
• Skewed Left: Tail extends to the left.

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Bell-Shaped Distributions

• Most individuals cluster around the mean.
• Normal Distribution: A special bell-shaped case.
  • Notation: $(N(\mu ,\sigma ))$.

68-95-99.7 Rule

• 68%: Within 1 standard deviation of the mean.
• 95%: Within 2 standard deviations.
• 99.7%: Within 3 standard deviations.

Example: Study Scores

• $(\mu =30)$, $(\sigma =7)$.
  • Middle 68%: 23 to 37.
  • Middle 95%: 16 to 44.
  • 99.7%: 9 to 51.
  • Scores > 23: ~84% (using Normal distribution properties).

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Standard Normal Distribution

• z-Score: Measures how many standard deviations a value is from the mean.
  • Example: Height of Malaysian women $((\mu =158.2)$ cm, $(\sigma =6.24)cm)$.
    • For 164.5 cm: $(z=\frac{164.5-158.2}{6.24}\approx 1.01)$.

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Quantile-Quantile Plot (Q-Q Plot)

• Purpose: Check if sample data conforms to a hypothesized distribution (e.g., Normal).
• Method: Plot empirical quantiles against theoretical quantiles.
  • Points should lie close to a straight line for Normal data.

Example in R

diameters <- c(1.24, 1.36, 1.31, 1.20, 1.39, 1.35, 1.41, 1.58, 1.49, 1.32)  
qqnorm(diameters, main = "QQ Plot of Soil Grain Diameters")  
qqline(diameters, col = "blue")