Descriptive Statistics, Central Limit Theorem & Dispersion
“Mean, median, mode, standard deviation, skewness, sample mean variance, and the Central Limit Theorem.”
1.Descriptive Measures and the Power of the Central Limit Theorem
§ 5.1.Statistics summarizes empirical observations and quantifies inference precision.
Comparison of Central Tendency Measures:
- Mean: Mathematically tractable, utilizes all data points, highly sensitive to outliers.
- Median: Divides data into equal halves, resilient to outliers (e.g., income, network latency).
- Mode: Only measure applicable to categorical/nominal data.
Skewness:
- Symmetric (Normal): Mean = Median = Mode.
- Positively Skewed (Right-tailed): Mode < Median < Mean.
- Negatively Skewed (Left-tailed): Mean < Median < Mode.
The Central Limit Theorem (CLT):
Regardless of the underlying distribution of the independent and identically distributed (i.i.d.) population (whether uniform, exponential, or skewed):
The distribution of the standardized sample mean converges to the standard normal distribution as sample size n grows large.
For the sum S_n: S_n ~ N(nmu, nsigma^2). For the average X_bar: X_bar ~ N(mu, sigma^2 / n). Always double-check whether the exam question asks for the probability regarding the sum of items or the average of items.
2. Worked Problem: Central Limit Theorem on Packet Transmission
Standard error is sigma / sqrt(n), do not divide by n.
Problem [Original Practice Problem · Modeled on GATE Pattern]:
The latency of packets through a router has an unknown distribution with mean mu = 20 microseconds and standard deviation sigma = 8 microseconds.
A sample of 64 packets is observed independently.
Using the Central Limit Theorem, find the probability that the sample mean latency exceeds 21.5 microseconds. (Given: Area under standard normal curve from 0 to Z=1.5 is 0.4332).
Step 1: Identify sample size and parameter values:
n = 64, mu = 20, sigma = 8.
Step 2: Compute mean and standard error of sample mean X_bar:
E[X_bar] = mu = 20 microseconds.
sigma_{X_bar} = sigma / sqrt(n) = 8 / sqrt(64) = 8 / 8 = 1.0 microsecond.
Step 3: Calculate the Z-score for X_bar = 21.5:
Z = (X_bar - mu) / sigma_{X_bar} = (21.5 - 20) / 1.0 = 1.5.
Step 4: Compute probability P(X_bar > 21.5):
P(X_bar > 21.5) = P(Z > 1.5) = 0.5 - P(0 < Z <= 1.5)
P(Z > 1.5) = 0.5000 - 0.4332 = 0.0668 (6.68%).
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