What quantitative data analysis is

• Quantitative data analysis is the systematic application of statistical and logical techniques to describe, summarise, and compare numerical data (Babbie)
• Its purpose is to uncover patterns, relationships, and trends that no one can see by leafing through individual questionnaires
• It sits at stage five of the research process — after design, measurement, sampling, and collection — and inherits every decision made before it
• Analysis cannot rescue bad measurement or a biased sample; it can only make the most of the data it is given

The Meridian staff survey

• Our running dataset for today: meridian-staff-survey.csv — 84 employees across the company
• Nine variables per respondent:
  • Who they are — division (Retail / Ecommerce / HQ), contract type, tenure in months, age group
  • What they think — engagement, pay satisfaction, and schedule satisfaction, each on a 1–5 scale
  • What they intend — do they intend to leave in the next year? (Yes / No)
• The board’s question is simple to state: who is at risk of leaving, and what distinguishes them?
• Every technique today will be a step towards answering it

The data matrix

• Quantitative data live in a data matrix: one row per case (here, one employee), one column per variable
• Each cell holds exactly one value — the case’s score on that variable
• This rectangle is the universal format: every spreadsheet, statistics package, and survey platform assumes it
• The matrix is only usable alongside its codebook — the document that records, for every variable, its name, its question wording, and what each value means
• Without a codebook, a matrix is just numbers; six months later, even its author cannot read it

Coding the data

• Coding assigns a value to every answer so it can be stored and counted
• Closed questions largely code themselves: the questionnaire’s response options become the values (Yes/No; 1–5 on a Likert item)
• Decisions still have to be made, and recorded in the codebook:
  • numeric codes or text labels (1/2 vs Full-time/Part-time)
  • how to treat “don’t know” and refused answers
  • how to band continuous answers if bands are wanted (age → age groups)
• The golden rule: code consistently and document everything — an undocumented coding decision is a future error

Cleaning the data

• Real data arrive dirty; cleaning finds and fixes errors before analysis, and it is not optional
• What to look for:
  • Impossible values (“wild codes”) — an engagement score of 7 on a 1–5 scale; a tenure of 480 months
  • Inconsistencies — an employee aged 18–24 with fifteen years of tenure
  • Duplicates — the same respondent entered twice
  • Missing values — and a policy for handling them, recorded openly
• The frequency table is your first cleaning tool: run one for every variable and any value outside the legal range shows itself
• Cleaning decisions are research decisions — log what you changed and why

Level of measurement constrains analysis

• Recall session 8: how a variable is measured fixes what arithmetic makes sense

• The scale carries downwards, not upwards: everything allowed for a nominal variable is allowed for an ordinal one, and so on
• A mean of division is nonsense; a mean of tenure_months is fine; a mean of a 1–5 rating is common practice but strictly a convenience — the intervals are not guaranteed equal
• First question of any analysis: what level is this variable?

Univariate description: the first look

• Univariate analysis describes one variable at a time — always the first step, never skipped
• It answers the descriptive questions: what values occur, how often, what is typical, how spread out?
• Three jobs at once:
  • Know your data — you cannot interpret a relationship between variables you have never looked at individually
  • Catch errors — impossible values surface here
  • Report the basics — the board’s first questions are univariate ones
• The tools: frequency distributions, measures of central tendency, measures of dispersion

Frequency distributions and percentages

• A frequency distribution lists each value of a variable with the number of cases taking it
• Convert counts to percentages so distributions can be compared across groups of different sizes — but always report the base N alongside
• “43% of Retail staff” means something very different when N = 60 than when N = 7

Central tendency: mode, median, mean

• Three answers to “what is the typical value?” — and level of measurement decides which are available

• The mode is crude but universal; the median is robust to extreme values; the mean uses all the information but is dragged around by outliers
• Reporting all three costs nothing and often tells a story by itself

When mean and median disagree

• If a distribution is symmetric, mean and median coincide; when they diverge, the distribution is skewed
• Mean well above median → a long right tail: a minority of very large values pulls the average up
• Classic right-skewed variables: income, firm size, hospital stays — and job tenure

• Rule of thumb: for skewed variables, lead with the median and mention the mean; for symmetric ones, the mean is fine

Dispersion: range and standard deviation

• Two distributions can share a mean and still be utterly different — central tendency without dispersion is half a description
• The range (maximum minus minimum) is the simplest measure, but it depends entirely on the two most extreme cases
• The standard deviation summarises, roughly, how far cases typically sit from the mean
  • small SD → cases cluster tightly around the mean; the mean represents them well
  • large SD → cases are spread widely; the mean represents them poorly
• No formula needed today — read the SD as a “typical distance from average”

Bivariate description: the question changes

• Bivariate analysis examines two variables together; the question shifts from what is typical? to do these vary together?
• This is where analysis meets the research question, which is almost always relational: does engagement differ by contract? does intention to leave depend on satisfaction?
• The tool depends on the levels of measurement of the pair:
  • two categorical variables → crosstab
  • one categorical, one numeric → compare group means
  • two numeric variables → correlation
• Association is a pattern in the data; explanation is a claim about the world — keep the two ideas apart (more shortly)

Crosstabs: two categorical variables

• A crosstab (contingency table) counts cases for every combination of two categorical variables

• Raw counts are honest but hard to compare — 18 is more than 13, yet there are nearly twice as many full-timers
• To compare fairly, we need percentages — and which way we percentage is the single most important decision in reading a crosstab

Percentaging a crosstab the right way

• Rule: percentage within the categories of the independent variable, then compare across them
• Here contract type is the (candidate) independent variable → percentage within each contract row:

• Now the comparison is fair: 44.8% of part-timers intend to leave against 32.7% of full-timers — a 12-point gap
• Percentaging the other way (“58% of leavers are full-time”) answers a different question — who makes up the leavers — and invites a wrong conclusion, because full-timers dominate simply by being more numerous

Comparing group means

• When the outcome is numeric (or a rating treated as numeric) and the grouping is categorical, compare the group means
• The logic mirrors the crosstab: split the cases by the independent variable, summarise the dependent one within each group, compare across

• A gap of 1.3 points on a 5-point scale is large; leavers are not slightly less engaged, they are a different population
• Always report group sizes with group means — a mean over 5 cases and a mean over 53 do not deserve equal confidence

Correlation, at the level of intuition

• For two numeric variables, correlation summarises how they move together in one number, r, between −1 and +1
  • positive — high values go with high values (tenure and age)
  • negative — high values go with low values (schedule satisfaction and intention to leave)
  • near zero — no linear pattern
• The size of r measures strength: ±0.1 is weak, ±0.3 moderate, ±0.5 strong (rough social-science conventions)
• Correlation captures only straight-line association, and a single number can hide very different patterns — never report an r for data you have not looked at

Reading and presenting tables

• A table is an argument; a good one can be read without the surrounding text. Every honest table has:
  • a title that states what, who, and when (“Intention to leave by contract type, Meridian staff survey, N = 84”)
  • labelled rows and columns, with units and scale ranges spelt out
  • the base N for every percentage
  • a source note, and a note of who was excluded
• Round sensibly — one decimal place is almost always enough; false precision (“32.727%”) signals naivety, not rigour
• State which way a crosstab is percentaged; make the 100% totals visible
• Reading others’ tables, run the same list as a checklist: no N, no labels, no source — no trust

How sure can we be? Sampling variability

• Our 84 respondents are a sample of Meridian’s ~450 staff; a different 84 would have produced slightly different numbers
• This is sampling variability: sample results dance around the population value from sample to sample
• Small samples dance more; large samples dance less — which is why base N matters so much
• Statistical inference asks: is the pattern in our sample strong enough that it is unlikely to be a fluke of who happened to respond?
• We need the idea, not the machinery — no formulas today

Significance and confidence, in plain words

• A confidence interval turns one number into an honest range: “36.9% intend to leave, give or take about 10 points” — the plausible values, given a sample this size
• A statistically significant difference is one too large to be plausibly explained by sampling variability alone — “probably not a fluke”
• Two warnings that save careers:
  • significance is not importance — with a huge sample, trivial differences become “significant”; with a tiny one, real differences fail to
  • the calculations assume a decent (probability) sample — no arithmetic repairs a biased one (recall session 8 on sampling)
• For this course: read significance claims as “how sure can we be?”, and always ask for the interval, not just the verdict

Pitfall 1: correlation is not causation

• An association between X and Y is compatible with at least four stories:
  • X causes Y
  • Y causes X (reverse causation)
  • Z causes both (a lurking third variable)
  • chance
• Description tells you that variables travel together; it cannot tell you why
• Establishing cause needs design — experiments, time order, controls (sessions 5 and 8) — not just a strong correlation

Pitfall 2: percentaging the wrong way

• The most common table error in real reports: percentaging within the dependent variable and reading the result as risk
• “58% of employees who intend to leave are full-time” → true, and almost inevitable, because 65% of the workforce is full-time
• The wrong-way percentage reflects group size, not group risk; the right-way percentage (32.7% vs 44.8%) reverses the apparent conclusion
• Defence: before reading any crosstab, find the 100% totals and ask what exactly is this a percentage of?
• If a claim compares groups, the percentages must be computed within those groups

Pitfall 3: tiny subgroups

• Percentages hide their base: “33% of HQ respondents under 25 intend to leave” may be one person out of three
• Small subgroups make wild percentages — one respondent changing their mind swings the figure by 33 points
• Rules of thumb:
  • always print the N beside any subgroup percentage
  • be sceptical of any percentage on fewer than ~30 cases; below ~10, report the raw count instead (“1 of 3”)
  • resist the urge to keep splitting until something dramatic appears — with enough slicing, noise always obliges

Tools of the trade

• For this course, a spreadsheet is enough: Excel or Google Sheets
  • COUNTIF / AVERAGE / MEDIAN for univariate work
  • pivot tables for frequency tables, crosstabs, and group means — the workhorse of today’s exercise
• One step further sit the statistics packages: jamovi (free, friendly), SPSS (the corporate and academic standard), R and Python (free, powerful, code-based)
• They automate the same logic you learn in a spreadsheet — significance tests, intervals, and models on top of the descriptions
• The tool is not the skill: knowing which table to build and how to read it transfers across every package

Conclusion

• Analysis starts before statistics: a coded, cleaned, documented data matrix, with the level of measurement of every variable known
• Univariate first — frequencies, percentages with their N, mode/median/mean chosen to fit the level, dispersion alongside
• Bivariate next — crosstabs percentaged within the independent variable, group means with group sizes, correlation read for direction and strength
• Inference, gently: sample numbers wobble; significance and confidence intervals measure how sure we can be, not how much it matters
• The three classic sins — causal claims from correlations, wrong-way percentages, tiny-base drama — are all avoidable by asking what is this a percentage of, and how many cases is it based on?
• In today’s exercise you will do all of this yourselves, on the Meridian survey, with a pivot table
• Questions and discussion are welcome

Exercise

Study guide