BUSINFO700 Chap.5 Univariate Analysis: Variables, Centre and Spread
Univariate Analysis: Variables, Centre and Spread
Univariate analysis describes one variable at a time, and this chapter follows the Week 3 lecture and lab in starting with the question that controls everything else: is the variable non-metric (binary, nominal or ordinal) or metric (interval or ratio)? Non-metric variables are summarised with frequency tables of counts, percentages and, where the categories are ordered, cumulative percentages.
Metric variables are summarised by their centre, using the mean, median and mode, by their spread, using the variance and standard deviation, and by their shape, using skewness and the course's cut-offs for slight and strong skew.
The chapter explains why business counts are often right skewed, how a left tail shows up when most values sit near a ceiling, why a mode of zero can be a genuine finding, and how z-scores place variables measured on different scales onto one standardised ruler. It separates the calculations that build understanding from the interpretation the short test questions ask for.
What this chapter covers
- 01
Non-metric and metric measurement levels
- 02
Frequency tables with percent and cumulative percent
- 03
Mean, median and mode on skewed data
- 04
Variance and standard deviation with n minus 1
- 05
Skewness and the slight and strong cut-offs
- 06
Z-scores for comparing different scales
Worked example · free
Compare one customer on two number lines with z-scores
- 1Write the z-score rule: z equals the observation minus the mean, divided by the standard deviation.
- 1Annual spend: z = (340 − 220) / 80 = 120 / 80 = 1.5 standard deviations above the mean.
- 1Orders per year: z = (11 − 6) / 2 = 5 / 2 = 2.5 standard deviations above the mean.
- 1Compare on the common scale: 2.5 is further from 0 than 1.5, so order frequency is the more unusual feature.
Key terms
- Measurement level
- The scale type of a variable, binary, nominal, ordinal, interval or ratio, which limits the analyses that are valid.
- Frequency table
- A summary listing each category of a variable with its count, its percentage and, for ordered categories, a cumulative percentage.
- Median
- The middle value once observations are sorted, so half lie below it and half above.
- Mode
- The value or category that occurs most often in a variable.
- Variance
- Average squared distance from the mean, computed with n minus 1 in the denominator for a sample.
Univariate Analysis: Variables, Centre and Spread FAQ
Why does the course insist on knowing the measurement level first?
The level determines which summaries make sense. A mean of postcodes or of income band codes is arithmetic on labels and tells you nothing. Non-metric variables get frequency tables, while metric variables get the mean, median, mode, standard deviation and skewness. The same rule later decides which bivariate test applies.
What does it mean when the mean is much larger than the median?
It usually signals a long right tail: a small number of very large values pull the mean up while the median stays with the typical case. Customer spending and transaction counts often behave this way. Report the median as the typical value and describe the tail separately.
Why build a frequency table before drawing a chart?
The table fixes the counts and percentages that any chart must show. The Week 3 lecture notes that a picture can say more than a thousand words but also raises the risk of distortion, for example through a truncated axis or slices that misstate shares. Check every bar or slice against the table before using the chart.
What skewness value counts as strongly skewed in this course?
The Week 3 slides treat a skewness above 1 as slightly right skewed and above 2 as strongly right skewed. Values below minus 1 and minus 2 mark slight and strong left skew in the same way. A value near zero indicates a roughly symmetric distribution.
Why would an analyst keep customers whose transaction count is zero?
Inactive customers are still part of the client base, so dropping them would overstate typical activity. The Week 3 lab checks this with the manager before deciding and keeps them. The general lesson is that whether a value belongs in the analysis is a business decision as well as a statistical one.
Assessment move
Make a habit of labelling every variable's measurement level before doing anything else; write it in the margin of every practice question. Then practise the three summaries in order: centre, spread and shape.
Working the support-ticket example by hand once is worthwhile because it shows what variance and skewness measure, but the practice test's short questions supply the summary figures and need no calculations, so most of your time should go on interpretation. Take sets of mean, median and mode, some right skewed and some left skewed, and say in one sentence what each implies about the customers.
For z-scores, invent two variables on different scales and place one person on both, then say which value is more unusual. Finally, practise answers of about thirty words that name the statistic, state what it implies about the distribution and add the business consequence, such as which centre a manager should quote.
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