Combined Data Analysis – pooled mean, variance, five‑number summary & box plot

1. What is “Combined Data Analysis”?

Combined data analysis (also called pooled or aggregate analysis) deals with merging two or more independent samples to obtain overall descriptive statistics. It is used when the underlying variable is the same (e.g., height, sales amount) but data are collected separately (different regions, time periods, or product lines).

Key concepts:

Note: When more than two groups are involved, extend the formulas by summing over all groups.

2. Computing the Combined Mean

Worked Example 1 (Exam style)

A sample of heights of 6 400 Indians has a mean of 67.85 inches, SD = 2.56 in. A sample of 1 600 British has a mean of 68.55 inches, SD = 2.52 in. Find the combined mean.

$${\bar{x}}_c=\frac{6400(67.85)+1600(68.55)}{6400+1600}=\frac{434240+109680}{8000}=\frac{543920}{8000}=67.99\text{ inches}$$

Interpretation: The overall average height of the combined sample (8 000 people) is 67.99 inches.

3. Combined Variance & Standard Deviation

Worked Example 2 (Unequal variances)

Two investment companies A and B each have 100 observations.
Company A: mean = 28 (000 Rs), variance = 25.
Company B: mean = 37 (000 Rs), variance = 36.
Compute the combined standard deviation.

1. Compute pooled variance (assuming equal population variance is not justified here, so we use the unbiased combined variance).

$$s_c^2=\frac{(100-1)25+(100-1)36+\frac{100\cdot 100}{200}(28-37{)}^2}{200-1}$$

$$=\frac{99\cdot 25+99\cdot 36+50\cdot 81}{199}=\frac{2475+3564+4050}{199}=\frac{10089}{199}=50.70$$

2. Combined standard deviation

$$s_c=\sqrt{50.70}=7.12\text{ (000 Rs)}$$

When to use pooled variance

If a preliminary test (e.g., Levene’s test) shows no significant difference in variances, the simpler pooled variance formula can be applied, saving one term in the numerator.

4. Five‑Number Summary & Box Plot

Worked Example 3

Data (in thousands of Rs):

$$39,26,15,8,70,11,45,60,20,32,52$$

1. Order the data

$$8,11,15,20,26,32,39,45,52,60,70$$

2. Minimum = 8, Maximum = 70

3. Median (Q2) – 6th value (since $n=11$) → 32

4. Lower quartile (Q1) – median of lower half (first 5 values) → median of {8,11,15,20,26} = 15

5. Upper quartile (Q3) – median of upper half (last 5 values) → median of {39,45,52,60,70} = 52

Five‑number summary: $\{8,15,32,52,70\}$

Shape comment: The median (32) is closer to Q1 (15) than to Q3 (52), indicating a slight right‑skew (longer upper tail).

5. Quartiles from Raw Data

Worked Example 4

Data:

$$78,12,11,43,88,11,0,10,7,62,12,21,30,95,78,13,98,9,89,12,5,9,89,12,5$$

1. Sort the data (n = 25).

2. Median (13th value) = 30

3. Lower quartile (Q1) = median of first 12 values = 11

4. Upper quartile (Q3) = median of last 12 values = 78

6. Coefficient of Variation (CV)

If the mean $\bar{x}=20$ and CV = 30 %, then

$$s=CV\times \bar{x}/100=0.30\times 20=6$$

Thus the standard deviation is 6.

7. Combining Regression Coefficients (Brief)

When two independent data sets share the same explanatory variable $X$ and we wish to fit a single regression line, we can pool the sums of squares:

$${\beta }_{1c}=\frac{{\sum }_{i=1}^{n_1}(x_i-{\bar{x}}_1)(y_i-{\bar{y}}_1)+{\sum }_{j=1}^{n_2}(x_j-{\bar{x}}_2)(y_j-{\bar{y}}_2)}{{\sum }_{i=1}^{n_1}(x_i-{\bar{x}}_1{)}^2+{\sum }_{j=1}^{n_2}(x_j-{\bar{x}}_2{)}^2}$$

$${\beta }_{0c}={\bar{y}}_c-{\beta }_{1c}{\bar{x}}_c$$

The exam rarely asks for full derivation; knowing the principle is enough.

8. Comparison of Methods

9. Applications in Business

10. In the real world

1. eSewa – The digital wallet aggregates transaction amounts from dozens of merchants each day. By calculating the combined mean transaction value, eSewa can set minimum top‑up thresholds and predict cash‑out requirements.

2. Daraz order queue – Daraz monitors order processing times for two major warehouses. Using the combined standard deviation, the logistics team estimates the overall variability in delivery time, allowing them to allocate extra staff during peak periods.

3. Ncell network usage – Ncell collects data‑usage (GB) from urban and rural towers. The coefficient of variation of the combined data tells the network engineers whether usage is uniformly spread (low CV) or highly variable (high CV), guiding capacity upgrades.

Worked example tie‑in: The height‑comparison problem (Indian vs. British) mirrors how a multinational retailer might compare average basket sizes across two countries before deciding on a unified pricing strategy.

11. Step‑by‑Step Procedure (Mermaid)

flowchart TD
  A["Start"] --> B["Compute Combined Mean"]
  B --> C["Compute Combined Variance"]
  C --> D["Compute Coefficient of Variation"]
  D --> E["Create Five‑Number Summary"]
  E --> F["Plot Box Plot"]
  F --> G["End"]

12. Summary of Key Formulas

$$\boxed{{\bar{x}}_c=\frac{\sum n_i{\bar{x}}_i}{\sum n_i}}$$

$$\boxed{s_c^2=\frac{\sum (n_i-1)s_i^2+\frac{\sum n_i{n}_j}{\sum n_i}({\bar{x}}_i-{\bar{x}}_j{)}^2}{\sum n_i-1}}$$

$$\boxed{CV=\frac{s}{\bar{x}}\times 100\%}$$

$$\boxed{\text{Five‑number summary}=\{\min ,Q_1,\text{Median},Q_3,\max \}}$$

In the real world

• eSewa aggregates transaction amounts from thousands of merchants daily; it uses the combined mean to set minimum top‑up thresholds and forecast cash‑out requirements.
• Daraz monitors order processing times from two major warehouses; the combined standard deviation helps the logistics team estimate overall delivery‑time variability and allocate staff during peak periods.
• Ncell collects data‑usage (GB) from urban and rural towers; the coefficient of variation of the combined data indicates whether usage is uniformly spread or highly variable, guiding capacity upgrades.

Exam tip

• Memorise the combined‑mean formula; it appears in every multi‑sample question.
• When the question gives means, SDs, and sample sizes, plug directly into the unbiased combined‑variance formula—don’t try to reconstruct raw data.
• For five‑number summary questions, always sort the data first; then locate positions using $\frac{n+1}{4}$ for Q1 and Q3 (round to nearest integer).
• Box‑plot drawing: mark min, Q1, median, Q3, max; whiskers extend to the smallest/largest values within 1.5 IQR. Outliers (if any) are plotted individually.
• Time management: allocate ~2 minutes for sorting, ~3 minutes for calculations, and ~1 minute for drawing the box plot.