Many MBA students arrive from engineering, commerce or the arts and find that statistics is the course that decides their GPA, and later the quality of their thesis or research project. The good news is that a handful of ideas carry you through most business research: describing data, estimating with confidence intervals, testing hypotheses and fitting a regression line. This guide walks through each one with a small dataset, the full hand calculation and the way results are reported in APA style.
Start With Descriptive Statistics
Before any test, describe your data. The essentials are:
- Mean \( \bar{x} \), the average value.
- Median, the middle value, which is better than the mean for skewed data such as salaries.
- Standard deviation \( s \), the typical distance of observations from the mean.
- Sample size \( n \), always reported.
The sample standard deviation divides by \( n - 1 \), not \( n \), because the sample mean has already used up one piece of information:
$$ s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}} $$
Worked Example 1: Branch Sales, a Confidence Interval and a t-Test
A retail chain records monthly sales (in PKR lakh) from a random sample of 8 branches:
12, 15, 11, 14, 18, 13, 16, 13
Management's old target was PKR 12 lakh per branch. Is average performance now different from 12?
Step 1: Mean
$$ \bar{x} = \frac{12 + 15 + 11 + 14 + 18 + 13 + 16 + 13}{8} = \frac{112}{8} = 14 $$
Step 2: Standard deviation
The deviations from 14 are −2, 1, −3, 0, 4, −1, 2, −1. Their squares are 4, 1, 9, 0, 16, 1, 4, 1, which sum to 36.
$$ s^2 = \frac{36}{8 - 1} = 5.143, \qquad s = \sqrt{5.143} \approx 2.268 $$
Step 3: Standard error
$$ SE = \frac{s}{\sqrt{n}} = \frac{2.268}{\sqrt{8}} = \frac{2.268}{2.828} \approx 0.802 $$
Step 4: 95% confidence interval
With \( n - 1 = 7 \) degrees of freedom, the critical t-value is \( t_{0.025,\,7} = 2.365 \).
$$ \bar{x} \pm t \times SE = 14 \pm 2.365 \times 0.802 = 14 \pm 1.896 $$
The interval is 12.10 to 15.90 lakh. We are 95% confident that the true mean branch sales lie in this range.
Step 5: One-sample t-test
Hypotheses: \( H_0: \mu = 12 \) and \( H_1: \mu \ne 12 \).
$$ t = \frac{\bar{x} - \mu_0}{SE} = \frac{14 - 12}{0.802} \approx 2.49 $$
With 7 degrees of freedom, the two-tailed p-value is about .041. Since \( p < .05 \), we reject \( H_0 \). Notice the agreement with Step 4: the value 12 lies just outside the confidence interval, which is exactly what a significant two-tailed test at the 5% level implies.
Step 6: Effect size
Statistical significance does not tell you how big the effect is. Cohen's \( d \) does:
$$ d = \frac{\bar{x} - \mu_0}{s} = \frac{2}{2.268} \approx 0.88 $$
By Cohen's conventions (0.2 small, 0.5 medium, 0.8 large), this is a large effect.
Reporting in APA style
A one-sample t-test showed that mean monthly branch sales (M = 14.00, SD = 2.27) were significantly higher than the previous target of 12 lakh, t(7) = 2.49, p = .041, d = 0.88, 95% CI [12.10, 15.90].
APA style italicises statistical symbols, gives degrees of freedom in brackets, and drops the leading zero for p-values because they cannot exceed 1.
Worked Example 2: Simple Linear Regression
A marketing manager wants to know how advertising spend relates to sales. Five months of data (both in PKR lakh):
| Month | Ad spend \( x \) | Sales \( y \) |
|---|---|---|
| 1 | 1 | 3 |
| 2 | 2 | 5 |
| 3 | 3 | 6 |
| 4 | 4 | 8 |
| 5 | 5 | 9 |
Step 1: Means
$$ \bar{x} = \frac{15}{5} = 3, \qquad \bar{y} = \frac{31}{5} = 6.2 $$
Step 2: Sums of squares and cross-products
The \( x \) deviations are −2, −1, 0, 1, 2 and the \( y \) deviations are −3.2, −1.2, −0.2, 1.8, 2.8.
$$ S_{xy} = (-2)(-3.2) + (-1)(-1.2) + 0 + (1)(1.8) + (2)(2.8) = 6.4 + 1.2 + 0 + 1.8 + 5.6 = 15 $$
$$ S_{xx} = 4 + 1 + 0 + 1 + 4 = 10 $$
$$ S_{yy} = 10.24 + 1.44 + 0.04 + 3.24 + 7.84 = 22.8 $$
Step 3: Slope and intercept
$$ b = \frac{S_{xy}}{S_{xx}} = \frac{15}{10} = 1.5, \qquad a = \bar{y} - b\bar{x} = 6.2 - 1.5(3) = 1.7 $$
The fitted line is \( \hat{y} = 1.7 + 1.5x \). Each extra lakh of advertising is associated with 1.5 lakh more sales.
Step 4: Correlation and \( R^2 \)
$$ r = \frac{S_{xy}}{\sqrt{S_{xx} S_{yy}}} = \frac{15}{\sqrt{10 \times 22.8}} = \frac{15}{15.10} \approx 0.993, \qquad R^2 \approx 0.987 $$
About 98.7% of the variation in sales is explained by advertising in this sample. The slope's standard error works out to 0.10, so \( t(3) = 1.5 / 0.10 = 15.0 \), \( p < .001 \).
Reporting in APA style
Simple linear regression indicated that advertising spend significantly predicted sales, b = 1.50, t(3) = 15.00, p < .001, and explained 98.7% of the variance in sales, R² = .99.
Caution: five observations is far too few for a real thesis, and a strong correlation does not prove that advertising causes sales. Seasonal demand might drive both.
Which Software?
- SPSS is common in Pakistani business schools for t-tests, ANOVA and regression through menus.
- R and Python are free and increasingly expected for research.
- Stata is popular in economics and finance.
- Excel handles descriptive statistics, t-tests and regression with the Analysis ToolPak.
The regression above takes two lines in R:
r
ads <- c(1, 2, 3, 4, 5); sales <- c(3, 5, 6, 8, 9)
summary(lm(sales ~ ads))
Choosing the Right Test
- Comparing one mean with a target: one-sample t-test.
- Comparing two independent groups (male vs female customers): independent-samples t-test.
- Comparing three or more groups: one-way ANOVA.
- Relationship between two categorical variables: chi-square test.
- Predicting one variable from others: regression.
Always check assumptions, such as approximate normality and equal variances, and say in your methodology chapter how you checked them.
Did You Know?
- The t-test was published in 1908 by William Sealy Gosset under the pen name "Student" while he worked for the Guinness brewery.
- The term "regression" comes from Francis Galton's 19th-century studies of how children's heights "regress" towards the average.
- A p-value of .05 is a convention, not a law of nature; many journals now ask for effect sizes and confidence intervals as well.
Key Takeaways
- Always report \( n \), mean and standard deviation before any inferential test.
- A 95% confidence interval and a two-tailed test at the 5% level always agree.
- Report effect sizes such as Cohen's \( d \) or \( R^2 \) alongside p-values.
- Follow APA format: italic symbols, degrees of freedom and no leading zero on p.
- Correlation and regression show association, not causation.
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