CDT502 Chap.2 Describing Distributions
Describing Distributions
This chapter reads a distribution through location, spread, shape and unusual observations. It compares mean, median, range, interquartile range, variance and standard deviation, then connects each measure to a decision rather than treating software output as a checklist. Practice covers skew, measurement scale, bin choices, standardisation and the difference between workload counts and population rates.
A centre is useful only after the distribution has been inspected. The mean uses every magnitude and acts as a balance point, so it responds strongly to extremes. The median uses order and resists large tail values. Neither is automatically superior: the right choice depends on whether the decision concerns total burden, an ordinary case or a rare but expensive failure. State that purpose when selecting the headline statistic.
Spread gives the centre operational meaning. The range is easy to compute but rests entirely on two observations. The interquartile range describes the middle half and travels naturally with the median. Standard deviation reflects squared distance from the mean and returns to the original unit, making it valuable when variability around a mean matters.
Always state whether a population or sample convention controls the variance denominator. Shape can reveal a mixture that no summary resolves. Multiple peaks may indicate distinct customer segments or operating states; skew can make an arithmetic average unrepresentative of an ordinary record; a gap can show a process threshold.
Histogram bins and density smoothing affect the displayed form, so compare settings and use dots when the dataset is small. Treat unusual values as questions. Verify entry, measurement unit and context, then run a labelled sensitivity comparison rather than deleting inconvenient observations. A genuine extreme can be the strongest evidence for contingency planning.
Finish by choosing the smallest set of summaries that preserves the decision-relevant contrast and explaining what the compression leaves out.
What this chapter covers
- 01
Frequency and relative frequency
- 02
Mean, median and mode
- 03
Range and interquartile range
- 04
Variance and standard deviation
- 05
Skew, bins and outliers
- 06
Measurement scales and standardisation
Worked example · free
Equal means, unequal operating risk
- 2Verify that both group means equal ten.
- 3Compare the squared deviations from each mean.
- 3Translate the spread into a staffing implication.
Key terms
- Interquartile Range
- The distance between the first and third quartiles, covering the middle half of ordered values.
- Variance
- The average squared distance from the mean under a stated population or sample convention.
- Z-score
- A value's signed distance from its group mean measured in standard deviations.
Describing Distributions FAQ
Why can two groups with the same mean need different decisions?
The mean describes a balance point, not variability. One group may be tightly concentrated around the mean while another alternates between low and high values. Staffing, inventory and risk decisions respond to that spread, so compare standard deviation, interquartile range and the full distribution before applying one rule.
Does an outlier always mean the data are wrong?
No. An outlier is an observation that deserves investigation, not an automatic deletion. Check entry, units and process context. If genuine, it may identify a rare but expensive failure. Report sensitivity to its inclusion and choose a robust summary when the ordinary pattern matters separately.
Why can histogram shape change when the data do not?
A histogram groups observations using a chosen bin width and starting point. Different boundaries allocate the same values to different bars and can merge or separate visible peaks. Publish or record the bin rule, and compare a dot plot or cumulative display when the conclusion depends on fine shape.
Exam move
For each dataset, predict whether the mean will sit above or below the median before calculating. Build a four-line description covering centre, spread, shape and unusual values. Change units or move one extreme and predict which summaries change. This turns formulas into directional checks that catch most exam arithmetic errors.
Maintain a small deck of distributions with the same centre but different spread, and with the same spread but different shape. Explain the operational consequence before revealing the calculations. Rebin one dataset three ways and describe which apparent features are stable. When software prints a long summary table, cross out every statistic that does not change the decision.
For categorical codes, say the measurement scale aloud before calculating anything. End each session by converting one summary back into a plain-language statement with its unit, period and excluded cases. The goal is selective description, not maximum output.
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