SOC613 Chap.4 Central Tendency, Dispersion and Visualisation
Central Tendency, Dispersion and Visualisation
Descriptive analysis asks what is typical, how much observations vary, what shape the distribution takes and whether unusual cases require investigation. This chapter links mode, median and mean to measurement level and sensitivity. It pairs range, interquartile range, variance and standard deviation with the different spread questions they answer. Binary means are reported as proportions.
Histograms, bar charts and box plots remain analytical tools because summary statistics alone cannot reveal skewness, multiple peaks, miscoding or influential tails. The mode identifies the largest frequency and remains available for nominal data. The median requires order and describes the halfway point without allowing a few extremes to dominate.
The mean uses every numerical value, which makes it informative and sensitive at the same time. Its interpretation must always carry the original unit. For a binary zero-one field, that interpretation changes form: the mean is the proportion in the category coded one. Spread is not an optional supplement to center.
The range shows the full span, the interquartile range describes the middle half, and variance with standard deviation expresses deviation around the mean. Each measure discards and retains different information. Visualisation lets the analyst see those consequences by revealing skew, clusters, gaps and unusual observations. An outlier is first a case to investigate, not an instruction to delete.
Units, coding, data-entry history and substantive plausibility must be checked before any exclusion. A defensible descriptive paragraph therefore names valid N, center, spread, shape and material extremes, then explains why those summaries suit the measurement level and distribution. This check is also protection for every later model, because unchecked input can still generate polished output.
What this chapter covers
- 01
Mode, median and mean
- 02
Binary means as proportions
- 03
Range and interquartile range
- 04
Variance and standard deviation
- 05
Skewness and distribution shape
- 06
Outliers and defensible visualisations
Worked example · free
Describe a skewed distribution
- 1Identify the right-skewed shape.
- 1Choose the median as primary center.
- 1Choose the interquartile range as primary spread.
- 1Inspect rather than automatically delete extremes.
- 1Interpret every value in hours.
Key terms
- Mode
- The category or value with the greatest frequency.
- Median
- The ordered value dividing observations into lower and upper halves.
- Interquartile Range
- The difference between the third and first quartiles.
- Standard Deviation
- A measure of spread around the mean expressed in the variable's unit.
- Skewness
- Asymmetry in a distribution, often described by the direction of its longer tail.
Central Tendency, Dispersion and Visualisation FAQ
Which measures of center fit each level?
The mode is available at every level, the median requires order, and the mean requires meaningful approximately equal intervals. Choose from substantive measurement first, then use distribution shape and the reporting goal to decide among valid options.
Does a mean above the median prove right skew?
It suggests that upper values may be pulling the mean upward, but the pair alone does not prove distribution shape. Inspect a histogram or box plot, verify extreme records and describe the tail with the visual evidence.
When is standard deviation a useful description?
It is useful for interval-ratio data when the mean is meaningful and observations are being described around that center. Always give the unit and examine shape. A standard deviation does not diagnose data quality or practical importance by itself.
What does the interquartile range show?
It is the distance from the first to the third quartile and therefore covers the middle half of ordered observations. Because it ignores the tails, it pairs naturally with the median for skewed distributions or plausible extremes.
Should every extreme value be removed?
No. Check coding, units, the source record and substantive plausibility before deciding. A genuine extreme case may contain important information. If exclusion is justified, state the rule and show how the result changes.
Exam move
For every variable, create a four-line profile: measurement level, valid N, center and spread, then add a sentence about shape and unusual observations. Select summaries before requesting them from software. Recalculate a small example by hand so the median position, deviations and interquartile range remain conceptually clear. Pair each numeric summary with an appropriate graph and check whether they tell the same story.
If they disagree, investigate coding, missing values, multiple groups or extreme cases. Practise writing complete descriptive paragraphs with units instead of listing output. For binary variables, translate the mean into the proportion coded one and name that category explicitly.
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