DSC-152 LAB: Descriptive Statistics and Probability Distributions

Focus: This practical involves creating various statistical graphs to visualize data distributions.

Charts to Master:

• Histogram: Used for continuous frequency distributions. Bars are adjacent. Can have equal or unequal class intervals (in which case, frequency density must be plotted).
• Frequency Polygon: A line graph connecting the midpoints of the tops of histogram bars. Can also be drawn without a histogram.
• Bar Diagram (or Bar Chart): Used for discrete or categorical data. Bars are separated.
• Pie Chart: Shows the proportion of different categories in a circle. Calculate the angle for each component as (Component Value / Total Value) * 360°.
• Ogive Curves:
  • Less Than Ogive: Plots cumulative frequency (less than type) against the upper class boundary. It is a rising curve.
  • More Than Ogive: Plots cumulative frequency (more than type) against the lower class boundary. It is a falling curve.

Focus: Calculating the "center" or "average" of a dataset using different methods.

Formulas to Apply:

• Arithmetic Mean (μ or x-bar):
  • Ungrouped Data: Σx / n
  • Grouped Data: Σ(f * x) / N, where x = midpoint and N = Σf
• Median:
  • Ungrouped Data: The (n+1)/2-th value of the sorted data.
  • Grouped Data: L + [ (N/2 - C) / f ] * h
    • L = Lower boundary of median class
    • N = Total frequency
    • C = Cumulative frequency of the class before the median class
    • f = Frequency of the median class
    • h = Class width
• Mode:
  • Ungrouped Data: The most frequent value.
  • Grouped Data: L + [ (f₁ - f₀) / (2f₁ - f₀ - f₂) ] * h
    • L = Lower boundary of modal class
    • f₁ = Frequency of modal class
    • f₀ = Frequency of class before modal class
    • f₂ = Frequency of class after modal class
• Geometric Mean (G.M.) and Harmonic Mean (H.M.) for ungrouped data.

Focus: Calculating the "spread" or "variability" of a dataset.

Formulas to Apply:

• Range: Highest Value - Lowest Value
• Quartile Deviation (Semi-Interquartile Range): (Q₃ - Q₁) / 2
  • You must first calculate the first quartile (Q₁) and third quartile (Q₃) using a method similar to the median.
• Mean Deviation:
  • About Mean: Σ |x - mean| / n
  • About Median: Σ |x - median| / n
• Standard Deviation (σ): The square root of Variance.
• Variance (σ²):
  • Ungrouped Data: [ Σ(x - mean)² ] / n OR [ Σx² / n ] - (mean)²
  • Grouped Data: [ Σf(x - mean)² ] / N OR [ Σ(f * x²) / N ] - (mean)²

Focus: Combining statistics from two or more groups and comparing their variability.

Formulas to Apply:

• Combined Mean (x-bar₁₂):
  x-bar₁₂ = (n₁ * x-bar₁ + n₂ * x-bar₂) / (n₁ + n₂)
• Combined Variance (σ²₁₂):
  σ²₁₂ = [ n₁(σ₁² + d₁²) + n₂(σ₂² + d₂²) ] / (n₁ + n₂)
  • d₁ = x-bar₁ - x-bar₁₂
  • d₂ = x-bar₂ - x-bar₁₂
• Coefficient of Variation (C.V.):
  C.V. = (σ / |mean|) * 100
  • This is a relative measure of dispersion, expressed as a percentage. It is used to compare the variability of two datasets, even if their means are different.

Focus: Describing the shape of the distribution.

Formulas to Apply:

• Raw Moments (μ'ᵣ): (Moments about origin)
  • μ'₁ = Σ(f * x) / N = Mean
  • μ'₂ = Σ(f * x²) / N
• Central Moments (μᵣ): (Moments about mean)
  • μ₁ = 0 (always)
  • μ₂ = μ'₂ - (μ'₁)² = Variance
  • μ₃ = μ'₃ - 3μ'₂μ'₁ + 2(μ'₁)³
  • μ₄ = μ'₄ - 4μ'₃μ'₁ + 6μ'₂(μ'₁)² - 3(μ'₁)⁴
• Karl Pearson's Coefficient of Skewness (Sk):
  Sk = (Mean - Mode) / σ OR Sk = 3 * (Mean - Median) / σ
  • If Sk > 0, positively skewed (right tail).
  • If Sk < 0, negatively skewed (left tail).
  • If Sk = 0, symmetric.
• Moment Coefficient of Skewness (β₁):
  β₁ = μ₃² / μ₂³
  • β₁ = 0 for a symmetric distribution.
• Moment Coefficient of Kurtosis (β₂):
  β₂ = μ₄ / μ₂²
  • If β₂ = 3, Mesokurtic (Normal curve).
  • If β₂ > 3, Leptokurtic (More peaked than normal).
  • If β₂ < 3, Platykurtic (Flatter than normal).

Focus: Using the Principle of Least Squares to find the "best fit" curve for a set of (x, y) data points.

Curves to Fit:

• Straight Line (Polynomial of degree 1): y = a + bx
  • Normal Equations:
    Σy = na + bΣx
    Σxy = aΣx + bΣx²
  • Solve this 2x2 system of equations for 'a' and 'b'.
• Parabola (Polynomial of degree 2): y = a + bx + cx²
  • Normal Equations:
    Σy = na + bΣx + cΣx²
    Σxy = aΣx + bΣx² + cΣx³
    Σx²y = aΣx² + bΣx³ + cΣx⁴
  • Solve this 3x3 system for 'a', 'b', and 'c'.
• Exponential Curve: y = abˣ
  • Transform first: Take log on both sides.
    log(y) = log(a) + x * log(b)
  • Let Y = log(y), A = log(a), B = log(b).
  • The equation becomes Y = A + Bx, which is a straight line.
  • Fit this line using the normal equations for a straight line:
    ΣY = nA + BΣx
    ΣxY = AΣx + BΣx²
  • Solve for A and B, then find the original parameters: a = antilog(A) and b = antilog(B).

Focus: Calculating the strength and direction of the linear relationship between two quantitative variables (x, y).

Formula (r):

Computational Formula:

Properties of 'r':

• -1 ≤ r ≤ +1
• r = +1: Perfect positive linear correlation.
• r = -1: Perfect negative linear correlation.
• r = 0: No linear correlation.

Focus: Calculating Karl Pearson's 'r' when the data is given in a bivariate frequency table (a "correlation table").

Formula:

• This is the same formula as Practical 7, but using coded values (u, v) and frequencies (f).
• u = (x - A) / h (step-deviation for x)
• v = (y - B) / k (step-deviation for y)
• fᵤ and fᵥ are the marginal frequencies.
• Σ(f * u * v) is the sum of the product of frequencies and their respective u,v values from the cells of the table.

Focus: Finding the "best fit" straight line (y = a + bx) to predict one variable from another.

Two Lines of Regression:

1. Regression Line of Y on X: (Used to predict Y if X is known)
   (y - y-bar) = bᵧₓ * (x - x-bar)
   • bᵧₓ is the regression coefficient of Y on X.
   • bᵧₓ = Cov(x, y) / σₓ² = r * (σᵧ / σₓ)
2. Regression Line of X on Y: (Used to predict X if Y is known)
   (x - x-bar) = bₓᵧ * (y - y-bar)
   • bₓᵧ is the regression coefficient of X on Y.
   • bₓᵧ = Cov(x, y) / σᵧ² = r * (σₓ / σᵧ)

Focus: Calculating the correlation between two variables when the data is ranked (ordinal). It measures the strength of a monotonic relationship (not just linear).

Formulas:

• Case 1: No Ties in Ranks
  R = 1 - [ (6 * Σd²) / (n³ - n) ]
  • d = R₁ - R₂ (Difference between the ranks for each observation)
  • n = Number of pairs
• Case 2: With Ties in Ranks
  • If a tie occurs, assign the average rank to all tied items.
  • Calculate Correction Factors (C.F.) for each tie:
    C.F. = (m³ - m) / 12, where 'm' is the number of items in a tie.
  • Calculate Σ(C.F.ₓ) for all ties in the x-variable and Σ(C.F.ᵧ) for all ties in the y-variable.
  • Use the general formula (which is just Karl Pearson's 'r' applied to ranks).

Focus: Given an observed frequency distribution (Oᵢ), find the expected frequencies (Eᵢ) according to a theoretical distribution (e.g., Binomial, Poisson).

General Procedure:

1. Estimate Parameters:
   • Binomial (n, p): 'n' is usually given. Estimate 'p' by setting the observed mean (x-bar) equal to the theoretical mean (np).
     x-bar = np => p = x-bar / n
   • Poisson (λ): Estimate 'λ' by setting the observed mean (x-bar) equal to the theoretical mean (λ).
     λ = x-bar
2. Calculate Probabilities:
   • Binomial: Use the p.m.f. P(x) = C(n, x)pˣ(1-p)ⁿ⁻ˣ to find P(0), P(1), P(2), ... P(n).
   • Poisson: Use the p.m.f. P(x) = (e⁻ˡᵃᵐᵇᵈᵃ * λˣ) / x! to find P(0), P(1), P(2), ...
   • Tip for Poisson: Use the recurrence relation: P(x) = P(x-1) * (λ / x). First, calculate P(0) = e⁻ˡᵃᵐᵇᵈᵃ, then P(1) = P(0)*(λ/1), P(2) = P(1)*(λ/2), etc.
3. Calculate Expected Frequencies (Eᵢ):
   Eᵢ = N * P(x)
   • Where N is the total observed frequency (N = ΣOᵢ).
4. Compare: Create a table of Observed Frequencies (Oᵢ) and Expected Frequencies (Eᵢ) to see how good the fit is. (Later, a Chi-Square Goodness-of-Fit test is used).

Focus: Using the properties of the Normal curve to find probabilities and fit it to data.

Practical 15: Applications

Focus: Using the properties of the Normal curve to find probabilities and fit it to data.

Practical 15: Applications

• This involves word problems like "Given X ~ N(μ, σ²), find P(a < X < b)."
• Procedure:
  1. Standardize the values 'a' and 'b' using the Z-formula:
     Z = (X - μ) / σ
  2. Z₁ = (a - μ) / σ
  3. Z₂ = (b - μ) / σ
  4. Find P(Z₁ < Z < Z₂) by looking up the areas for Z₁ and Z₂ in the Standard Normal (Z) table and subtracting them.

Practical 16: Fitting

• This is similar to fitting discrete distributions, but for continuous data.
• Procedure:
  1. Calculate the observed mean (x-bar) and standard deviation (σ) from the given frequency distribution. These are your estimates for μ and σ.
  2. For each class boundary (x), calculate the Z-score:
     Z = (x - μ) / σ
  3. Using the Z-table, find the cumulative area from -∞ up to each Z-score (P(Z ≤ z)).
  4. Find the area (probability) within each class interval by subtracting the cumulative areas of its boundaries.
     P(a < X < b) = P(Z < Z₂) - P(Z < Z₁)
  5. Calculate the Expected Frequency for that class:
     Eᵢ = N * P(a < X < b)
  6. Compare the Oᵢ and Eᵢ values.