Business Analytics
Deciding from evidence instead of instinct.
Never before has there been so much data — and so many confident conclusions drawn badly from it. This module is the discipline that stops that: describe a data set honestly, quantify how two things move together without claiming one causes the other, and say how much of what you see in a sample you are entitled to believe about the population.
The question a manager brings to it: What is the data actually telling us?
What it lets you do
- Read a distribution's shape, centre, spread and outliers before computing anything
- Pick mean, median or mode on purpose rather than by habit
- Quantify a relationship with correlation — and resist reading cause into it
- Turn a sample into a confidence interval, and state what that interval does and does not promise
Before any of it
Start with the question
The discipline does not begin with a spreadsheet. It begins with articulating, out loud and in one sentence, the question you are trying to answer. Everything downstream — which data you gather, how you group it, which graph you draw — is downstream of that sentence, and skipping it is how people end up with beautiful charts of the wrong thing.
After that, organise the data into categories that let you compare within and across groups, and draw it before you compute anything. Graphs reveal patterns, trends and outliers that summary statistics quietly average away.
Shape first
The histogram
A histogram cuts the range into bins and counts what falls into each. It is the fastest way to see three things at once: where the data piles up, whether it leans to one side, and whether anything is sitting far out on its own.
Sixty opening weekends, one histogram
The x-axis is a range of box office; the y-axis is how many films landed in that range. Bin width is a choice you make, and it decides what you are able to see.
Wide bins simplify and hide; narrow bins get so sparse the pattern disappears.
Films
60
Mean
$45.2M
Median
$19.3M
Skewness
1.12
skewed right
- Films per range
- Mean
- Median
The long right tail drags the mean well above the median — that is what “skewed right” looks like in practice. Pull the one blockbuster out and watch which of the two moves.
Show the numbers
| Range | Films |
|---|---|
| $0.0M – $20.0M | 31 |
| $20.0M – $40.0M | 12 |
| $40.0M – $60.0M | 5 |
| $60.0M – $80.0M | 3 |
| $80.0M – $100.0M | 2 |
| $100.0M – $120.0M | 1 |
| $120.0M – $140.0M | 1 |
| $140.0M – $160.0M | 1 |
| $160.0M – $180.0M | 1 |
| $180.0M – $200.0M | 0 |
| $200.0M – $220.0M | 1 |
| $220.0M – $240.0M | 0 |
| $240.0M – $260.0M | 1 |
| $260.0M – $280.0M | 0 |
| $280.0M – $300.0M | 0 |
| $300.0M – $320.0M | 0 |
| $320.0M – $340.0M | 0 |
| $340.0M – $360.0M | 0 |
| $360.0M – $380.0M | 0 |
| $380.0M – $400.0M | 0 |
| $400.0M – $420.0M | 1 |
- Bin width is a judgement, not a default
- Wide bins simplify the picture and can hide the trend you were looking for. Very narrow bins drop the count per bin so low that the pattern dissolves into noise. The right width depends on what you are trying to learn.
- Skewness is asymmetry, measured
- A long right tail pulls the mean above the median — skewed right. A long left tail does the opposite. The gap between mean and median is the quickest read on which way a distribution leans.
- Investigate an outlier before you touch it
- It might be a data-entry error, a different unit, or the single most interesting observation in the set. Leave it, correct it, or remove it — but only after you know which of those it is. Never quietly delete one.
Centre and spread
Summarising a data set in numbers
Three numbers describe the centre. The mean is the sum divided by the count, so every value tugs on it. The median is the middle value, so only position matters. The mode is whatever occurs most often, and a set can have more than one.
Spread needs its own numbers: the range, the variance, and the standard deviation — which is the square root of the variance, and the one you actually quote, because it comes back in the same units as the data.
Same average, different story
Mean tells you where the middle is. Range, standard deviation and the coefficient of variation tell you whether the middle means anything.
Steady Ltd · mean
24.5
Spiky Inc · mean
24.5
Steady Ltd · CV
0.059
σ 1.4
Spiky Inc · CV
0.585
σ 14.3
- Steady Ltd
- Spiky Inc
Steady and Spiky have almost the same mean and almost nothing else in common. Enterprise has a much larger standard deviation than Steady in raw numbers — but once you divide by the mean, the coefficient of variation says it is actually the more consistent of the two. That is why you compare spread across different scales with CV, not σ.
Show the numbers
| Team | Mean | Range | Std deviation | Coefficient of variation |
|---|---|---|---|---|
| Steady Ltd | 24.5 | 5 | 1.45 | 0.059 |
| Spiky Inc | 24.5 | 41 | 14.34 | 0.585 |
| Enterprise Co | 200.7 | 43 | 12.55 | 0.063 |
Two variables at once
Scatter plots and correlation
Plot two variables against each other and the relationship, if there is one, shows itself. Correlation puts a number on how tightly the cloud hugs a straight line: +1 is a perfect upward line, −1 a perfect downward one, 0 no line at all.
Correlation, and what it refuses to tell you
A single number between −1 and +1 for how tightly two variables track a straight line. Switch data sets and watch where it stops being useful.
Correlation r
0.934
strong
Observations
40
Mean Ad spend ($k)
55.0
Mean Revenue ($k)
128.2
A strong positive linear relationship. Correlation near +1 means the points hug an upward line — it does not mean the ads caused the revenue.
Show the numbers
| Ad spend ($k) | Revenue ($k) |
|---|---|
| 10.0 | 69.8 |
| 12.3 | 72.1 |
| 14.6 | 18.1 |
| 16.9 | 81.5 |
| 19.2 | 53.1 |
| 21.5 | 66.7 |
| 23.8 | 38.8 |
| 26.2 | 65.9 |
| 28.5 | 63.3 |
| 30.8 | 63.6 |
| 33.1 | 78.4 |
| 35.4 | 103.3 |
- Near zero does not mean “no relationship”
- It means no *linear* relationship. A clean U-shape has a correlation of roughly zero and an obvious pattern. Plotting the data is what catches this; r never will.
- Outliers move r more than you expect
- A single extreme point can inflate or collapse a correlation calculated over dozens of well-behaved ones. Check whether your r survives the removal of any one observation.
- Time series vs cross-sectional
- When one of the variables is time, you have a time series. A snapshot across many groups at one moment is cross-sectional data. They answer different questions and mixing them up is a classic error.
The famous warning
Correlation is not causation
Everyone can recite it. The useful version is more specific: when two things move together, a third thing you did not plot is often moving both. That third thing is a hidden variable, and it is the reason the warning exists.
The third variable nobody plotted
Ice cream sales and beach rescues, one point per day across a summer. The correlation is strong and the causal story writes itself — wrongly.
Correlation, all days
0.96
looks like a smoking gun
Within cool days only
—
reveal to see
Within warm days only
—
reveal to see
Days observed
56
r ≈ 0.9. If you stop here you can argue that ice cream causes drownings, or that near-drownings make people crave ice cream. Both are nonsense, and nothing in the correlation coefficient can tell you so.
Show the numbers
| Slice | Days | Correlation |
|---|---|---|
| All days together | 56 | 0.957 |
| Cool days (under 26°C) | 28 | 0.822 |
| Warm days (26°C and up) | 28 | 0.866 |
From the few to the many
Samples and populations
You almost never get the whole population, so you take a sample and reason from it. The numerical properties of a population are called parameters — μ and σ. The ones you compute from a sample are statistics — x̄ and s. The entire game is using the second to say something defensible about the first.
Sample size does not depend on population size, which surprises most people. It depends on the accuracy you need and, for rare events, on how rare the thing you are trying to observe is.
- Make it representative
- Choose members randomly so every member of the population has an equal chance of being included. Randomisation is what makes the variables you are not measuring cancel out across groups, and it is the defence against selection bias.
- Phrase questions neutrally
- A leading question produces a biased sample of answers no matter how clean your sampling frame was.
- Chase the response rate
- A smaller sample with a high response rate usually beats a larger one with a low response rate, because non-response is rarely random.
- Type I error is the significance level
- A 10% significance level means a 10% chance of a type I error — finding an effect that is not there. Significance level is 1 minus the confidence level, and the two get confused constantly.
The shape that keeps showing up
The normal distribution
A symmetric curve fully described by two numbers: its mean and its standard deviation. Because its shape is fixed, any value can be restated as a z-score — how many standard deviations it sits from the mean — and from that you can read off a probability.
The normal distribution, and the three numbers worth memorising
Mean sets where the curve sits; standard deviation sets how wide it is. Those two numbers are the entire shape.
moves the curve sideways
sets how wide it is
z-score
1.71
standard deviations from the mean
Below this value
95.7%
Above it
4.3%
Within 2σ
170 – 310
the 95% range
- Within 1σ · 68%
- Within 2σ · 95%
- Within 3σ · 99.7%
- Your value
About 68% of the probability sits within one standard deviation of the mean, about 95% within two (1.96, to be exact), and about 99.7% within three. A z-score just restates a value as “this many standard deviations out”, which is what makes those three numbers portable to any normal curve at all.
Show the numbers
| Range | Covers | Share of the distribution |
|---|---|---|
| 1σ either side | 205 – 275 | 68% |
| 2σ either side | 170 – 310 | 95% |
| 3σ either side | 135 – 345 | 99.7% |
| z for 300 | 1.71 | 95.7% below |
The result everything rests on
The Central Limit Theorem
Take enough sufficiently large samples from any population and the means of those samples will be normally distributed — regardless of what the population itself looks like. The mean of that distribution of sample means equals the population mean, and its standard deviation is σ divided by the square root of the sample size.
That is what lets you take a single sample mean, place it on a normal curve you never measured, and draw a conclusion from it.
Draw enough samples and the means go normal — whatever the population looks like
The population below is violently skewed. The distribution of its sample means is not, and that is the Central Limit Theorem in one picture.
Samples drawn
0
Mean of sample means
—
population mean $45.2M
Spread of sample means
—
theory says $12.6M
Standard error σ/√n
$12.6M
σ $69M ÷ √30
The population — badly skewed
The sample means — heading for normal
- Population
- Sample means
- True population mean
Switch to n = 5 and draw fifty: the pile is wide and lumpy. Switch to n = 100 and draw fifty: it tightens into a narrow bell centred on the population mean. The width is σ/√n, so quadrupling the sample only halves the spread — precision gets expensive fast.
Show the numbers
| Measure | Predicted by theory | What we actually got |
|---|---|---|
| Mean of the sample means | $45.16M | — |
| Standard deviation of the sample means | $12.57M (σ/√n) | — |
| Samples drawn | — | 0 |
| Population σ | $68.87M | — |
Saying how sure you are
Confidence intervals
A sample mean is a point estimate and will essentially always be a little wrong. A confidence interval turns it into a range, built so that a stated share of intervals constructed this way contain the true population mean. The width depends on the confidence level, the spread of the data, and the sample size — and only two of those three are yours to choose.
60 samples, 60 intervals
Five hundred support tickets, and 60 attempts to estimate their mean response time from a sample. Each line is one sample's interval; the vertical rule is the true mean, which in real life you never get to see.
Bigger samples make every interval narrower.
More confidence buys you a wider, vaguer interval.
Intervals that caught it
59 of 60
98% — aiming for 95%
Intervals that missed
1
Average width
5.2 min
wider means less useful
True mean
50.6 min
unknowable in real life
- Interval contains the true mean
- Interval misses it
- True population mean
What a 95% confidence interval does not say. It does not say there is a 95% chance the true mean is inside the one interval you calculated. Your interval either contains it or it does not — you just cannot tell which. The 95% is a property of the procedure: draw samples this way over and over, and about 95% of the intervals you build will catch the true mean. Which is exactly what the 60 lines above are showing you.
Show the numbers
| Setting | Value |
|---|---|
| Confidence level | 95% |
| z used | 1.96 |
| Sample size n | 40 |
| Intervals drawn | 60 |
| Intervals containing the true mean | 59 (98%) |
| Average interval width | 5.2 min |
| True population mean | 50.59 min |
- Large samples (n ≥ 30) use z
- x̄ ± z·s/√n. CONFIDENCE.NORM returns the margin of error you add and subtract.
- Small samples (n < 30) use t
- The t-distribution is shorter and wider than the normal, which gives a wider and more conservative interval. CONFIDENCE.T is its margin of error.
- Intervals for proportions
- Convert the data to dummy 0/1 variables and proceed as normal — but first check that n·p̄ ≥ 5 and n(1 − p̄) ≥ 5. If either fails, the sample is too small and you need more data, not a different formula.
- Smaller samples, wider intervals
- More uncertainty has to show up somewhere, and it shows up as width. Raising the confidence level widens the interval too — you can be very confident about a very vague claim.
The toolkit
The Excel functions behind all of this
Every idea on this page has a one-line implementation. Arguments in square brackets are optional, and the functions ending in .S apply to samples rather than whole populations.
| Function | What it gives you |
|---|---|
=AVERAGE(range) | Mean. |
=MEDIAN(range) | Middle value — the 50th percentile. |
=MODE.SNGL(range) | Most frequent value. A set can have several modes. |
=AVERAGEIF(range, criteria, [average_range]) | Conditional mean: the average of the rows that meet a condition. |
=PERCENTILE.INC(array, k) | The k-th percentile, k as a decimal — 0.95 for the 95th. |
=VAR.S(range) · =STDEV.S(range) | Sample variance and standard deviation. |
=CORREL(array1, array2) | Correlation coefficient, −1 to +1. |
=NORM.DIST(x, mean, sd, TRUE) | Cumulative probability of being at or below x on a normal curve. |
=NORM.S.DIST(z, TRUE) | The same thing on the standard normal, taking a z instead of a value. |
=NORM.INV(probability, mean, sd) | The reverse: the value sitting at a given cumulative probability. |
=CONFIDENCE.NORM(alpha, sd, size) | Margin of error for a large sample. Alpha is 1 − confidence, so 0.05 for 95%. |
=CONFIDENCE.T(alpha, sd, size) | The same for a small sample (n < 30), using the wider t-distribution. |
The Data Analysis tool also builds histograms and the full Descriptive Statistics output table in one pass, which is faster than assembling the measures one formula at a time.
Recap
The module in ten lines
The short version, in the order you would actually use them on a real data set.
- Write the question down first
- Everything else is downstream of it, including which data you bother collecting.
- Plot before you compute
- Shape, trend and outliers are visible in a graph and invisible in a summary table.
- Bin width changes what you can see
- It is a choice you are making, so make it deliberately.
- Mean moves, median holds
- An extreme value drags the mean and leaves the median roughly where it was.
- Quote the standard deviation, compare with CV
- σ is in the data's units; the coefficient of variation is not, which is why it travels.
- r only measures a straight line
- Near zero rules out a line, not a relationship — and one outlier can rewrite it.
- Look for the variable you did not plot
- Most spurious correlations are a third variable driving both of the first two.
- Randomise, and chase the response rate
- Representativeness beats raw size, and non-response is almost never random.
- Sample means go normal even when the population does not
- That is the Central Limit Theorem, and it is what makes inference possible at all.
- A confidence interval describes the procedure
- Not the single interval in front of you. 95% of intervals built this way catch the true mean.