Statistics & Data Interpretation

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Measures of Central Tendency:

• Mean (average): sum of all values divided by number of values. For data 4, 7, 9, 12, 18: mean = (4+7+9+12+18)/5 = 50/5 = 10.
• Median: the middle value when data is arranged in order. For an odd number of values, it’s the middle one. For an even number, it’s the average of the two middle values.
• Mode: the value that appears most frequently. A dataset may have no mode (all values unique), one mode (unimodal), or two modes (bimodal).

Range: difference between largest and smallest value. Range of {4, 7, 9, 12, 18} = 18 − 4 = 14.

⚡ Exam tip: The mean is affected by extreme values (outliers), while the median is more robust. If salaries are 3, 4, 5, 6, 100 (millions), the median of 5 is more representative than the mean of 23.6.

⚡ Exam tip: For grouped data, the median lies in the median class: L + [(n/2 − c) / f] × w, where L = lower boundary of median class, c = cumulative frequency before median class, f = frequency of median class, w = class width.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Mean, Median, Mode — When to Use Each

The mean is best for symmetric distributions without outliers. The median is preferred for skewed distributions or data with extreme values. The mode is useful for categorical data (most common response: “strongly agree”) but less useful for continuous numerical data where values rarely repeat.

Calculating the Mean from Frequency Tables

For ungrouped data: x̄ = Σfx / Σf, where f is frequency.

Value (x)	Frequency (f)	f×x
3	4	12
5	7	35
8	3	24
Total	14	71

Mean = 71/14 ≈ 5.07

Grouped Data — Finding the Modal Class

For grouped data, the mode lies in the class with highest frequency. The exact mode cannot be found, only estimated: Mode ≈ L + [(f₁ − f₀) / ((f₁ − f₀) + (f₁ − f₂))] × w, where L = lower boundary of modal class, f₁ = frequency of modal class, f₀ = frequency before, f₂ = frequency after, w = class width.

Variance and Standard Deviation

Variance measures the spread of data around the mean. For a dataset: σ² = Σ(x − x̄)² / n for population variance, or use n−1 for sample variance (Bessel’s correction).

Easier computational formula: σ² = (Σx² / n) − x̄²

For data {2, 4, 6, 8}: Σx = 20, x̄ = 5, Σx² = 4+16+36+64 = 120. σ² = 120/4 − 25 = 30 − 25 = 5. σ = √5 ≈ 2.24.

Quartiles and the Interquartile Range

When data is ordered: Q1 is the 25th percentile, Q2 is the median (50th), Q3 is the 75th percentile. The interquartile range (IQR) = Q3 − Q1.

For the dataset {3, 5, 7, 9, 11, 13, 15}: Q1 = 5, median (Q2) = 9, Q3 = 13. IQR = 13 − 5 = 8.

Outliers are often defined as values below Q1 − 1.5×IQR or above Q3 + 1.5×IQR.

Common Mistakes to Avoid:

Mistake	Correct approach
Confusing mean with median	Mean uses all values; median is the middle value
Forgetting to multiply frequency × value in frequency tables	Always create the f×x column
Using class boundaries instead of midpoints for grouped mean	Class midpoint = (lower + upper) / 2
Reporting standard deviation as variance	Variance is σ² (units²); standard deviation is σ (same units as data)
Not ordering data before finding median	Always sort first

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

The Effect of Adding Data Points on Statistical Measures

Adding a new value changes the statistics. If we add 20 to {2, 4, 6, 8} (mean = 5): new sum = 20+20 = 40, new n = 5, new mean = 8. The median changes from 5 to 6 (arranged: 2, 4, 6, 8, 20 → median is 6).

If we add a value equal to the current mean, the mean stays unchanged. Adding the value 5 to {2, 4, 6, 8}: original sum = 20, new sum = 25, n = 5, new mean = 25/5 = 5. Correct.

Combined Mean

If two groups have means x̄₁ and x̄₂ with sizes n₁ and n₂, the combined mean is: x̄ = (n₁x̄₁ + n₂x̄₂) / (n₁ + n₂).

Class 1 (30 students) has mean 72, Class 2 (20 students) has mean 80. Combined mean = (30×72 + 20×80) / 50 = (2160 + 1600) / 50 = 3760/50 = 75.2.

Skewness

A distribution is symmetric if mean ≈ median. Negatively skewed (left-skewed): mean < median (tail extends left). Positively skewed (right-skewed): mean > median (tail extends right).

Income distributions are typically positively skewed — a few high earners pull the mean up while most people earn below the mean. The median is often a better measure of “typical” income.

Standard Error and Confidence Intervals

For a sample of size n drawn from a population with standard deviation σ, the standard error of the mean is σ/√n. If σ = 15 and n = 100, SE = 15/10 = 1.5. A 95% confidence interval for the population mean is x̄ ± 1.96 × SE.

Coefficient of Variation

CV = (σ / x̄) × 100%. This normalises the standard deviation, allowing comparison of variability between datasets with different scales. Investment A: mean = 50, SD = 5, CV = 10%. Investment B: mean = 100, SD = 7, CV = 7%. Despite higher absolute volatility, B has relatively lower risk.

Normal Distribution Basics

In a normal distribution: approximately 68% of data falls within 1 SD of the mean, 95% within 2 SD, 99.7% within 3 SD. If exam scores are normally distributed with mean 65 and SD 10, about 68% of students score between 55 and 75.

Z-score = (x − x̄) / σ. A score of 80 has z-score = (80 − 65)/10 = 1.5, meaning it’s 1.5 standard deviations above the mean.

Historical Context

The word “statistics” comes from the Latin statisticum (matters of the state). John Graunt (1662) analysed London death records, pioneering demography. Carl Friedrich Gauss (1777-1855) developed the normal distribution curve. Francis Galton (1822-1911) coined “regression to the mean” and developed standard deviation. The phrase “lies, damned lies, and statistics” was popularised by Mark Twain (attributing it to British Prime Minister Benjamin Disraeli).

Exam Pattern Analysis

UI statistics questions commonly:

1. Calculate mean, median, or mode from raw data or frequency tables
2. Compare two datasets using mean and range (or standard deviation)
3. Find the median class and compute median from grouped data
4. Use statistical measures in word problems (average speed, weighted averages)
5. Interpret charts and graphs requiring statistical calculations

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