7.2 Descriptive Statistics: Presentation, Central Tendency, Variability and Standard Scores
Key Takeaways
- TOS competencies A.1.4, A.1.5 and A.1.6 cover tabular and graphical presentation, measures of central tendency, variability, score transformations and the normal distribution.
- The mean is sensitive to extreme values, the median is resistant, and the mode is the only central tendency valid for nominal data.
- Range, variance and standard deviation describe dispersion; two distributions with the same mean can differ entirely in spread.
- In a normal distribution approximately 68, 95 and 99.7 percent of cases fall within one, two and three standard deviations of the mean.
- A z-score expresses how many standard deviations a raw score lies from the mean and makes scores from different scales comparable.
7.2 Descriptive Statistics: Presentation, Central Tendency, Variability and Standard Scores
Blueprint anchor. TOS competencies A.1.4 ("illustrate the application of statistical functions, the tabular and graphical presentations and the measures of central tendency"), A.1.5 ("determine the ways of achieving variability and score transformations") and A.1.6 ("utilize the normal and other distribution and standard scores") sit inside the 10-item Social Work Statistics topic.
1. Presenting Welfare Data
| Method | Best used for | Caution |
|---|---|---|
| Textual | One or two figures inside a narrative | Becomes unreadable beyond a few numbers |
| Tabular (frequency distribution) | Exact values across categories | Needs clear title, column heads and total |
| Bar graph | Comparing nominal or ordinal categories | Bars separated; do not imply continuity |
| Histogram | Distribution of a continuous variable in class intervals | Bars adjoin; interval width must be uniform |
| Pie chart | Parts of a single whole | Useless beyond about six slices; parts must sum to the whole |
| Line graph | Trend over time | Requires ordered time on the horizontal axis |
| Scatter plot | Relationship between two continuous variables | Shows association, never causation |
A well-formed frequency distribution reports the class, the frequency, the percentage and the cumulative percentage. In welfare reporting, always add the base: "68 percent (n = 204 of 340)".
2. Measures of Central Tendency
| Measure | Definition | Strength | Weakness | Valid for |
|---|---|---|---|---|
| Mean | Sum of values divided by the number of values | Uses all data; basis of most inferential tests | Distorted by extreme values | Interval and ratio |
| Median | Middle value when data are ordered | Resistant to extremes; best for skewed income data | Ignores the magnitude of other values | Ordinal, interval, ratio |
| Mode | Most frequently occurring value | Only measure valid for categories | May not exist or may be multiple | All levels |
Worked example. Monthly incomes of seven assisted households, in pesos: 4,000; 5,000; 5,000; 6,000; 7,000; 8,000; 60,000.
- Mean = 95,000 ÷ 7 ≈ 13,571
- Median = the fourth ordered value = 6,000
- Mode = 5,000
The mean of ₱13,571 describes none of these households. For skewed distributions — income, assistance amounts, length of stay in residential care — the median is the honest summary. This is a standing rule in welfare reporting.
3. Measures of Variability
Two barangays can both report a mean household income of ₱12,000 while one is uniformly modest and the other contains extreme wealth alongside destitution. Dispersion measures capture that difference.
| Measure | Definition | Interpretation |
|---|---|---|
| Range | Highest value minus lowest value | Crude; determined entirely by two cases |
| Interquartile range | Spread of the middle 50 percent | Resistant to outliers; pairs with the median |
| Variance | Mean of the squared deviations from the mean | In squared units; used in ANOVA |
| Standard deviation | Square root of the variance | Same units as the data; the standard dispersion measure |
| Coefficient of variation | Standard deviation relative to the mean | Compares spread across variables with different units |
Ways of achieving or reporting variability, in the TOS phrasing, means the deliberate use of these measures — and of grouping, stratification and disaggregation — so that averages do not conceal inequality. A municipal report giving only means, without dispersion or disaggregation, is professionally incomplete.
4. The Normal Distribution
The normal curve is symmetric and bell-shaped, with the mean, median and mode coinciding at the centre. Its defining empirical rule:
| Interval around the mean | Approximate proportion of cases |
|---|---|
| ± 1 standard deviation | 68 percent |
| ± 2 standard deviations | 95 percent |
| ± 3 standard deviations | 99.7 percent |
Skewness describes asymmetry. In a positively (right) skewed distribution the tail extends to the right and the mean exceeds the median — the standard shape of income data. In a negatively (left) skewed distribution the tail extends to the left and the mean falls below the median.
5. Score Transformations and Standard Scores
A standard score (z-score) expresses a raw score as the number of standard deviations it lies from the mean:
- A z of 0 is exactly at the mean; +1.0 is one standard deviation above; −1.5 is one and a half below.
- Because z removes the original units, scores from different instruments become comparable.
Worked example. A social work student scores 78 on a field-performance rating with mean 70 and standard deviation 8, and 85 on a written examination with mean 80 and standard deviation 10.
- Field: z = (78 − 70) ÷ 8 = +1.00
- Written: z = (85 − 80) ÷ 10 = +0.50
Despite the lower raw score, the student performed relatively better in field practice. Raw score comparison would have reversed the conclusion — the reason standard scores exist.
Percentile rank gives the percentage of cases at or below a score. A score at the 75th percentile exceeds three quarters of the group. Percentile ranks are ordinal: the distance between the 50th and 60th percentile is not the same amount of the underlying trait as between the 85th and 95th.
6. Worked Practice Application
A regional office compares two residential care facilities on length of stay for children awaiting placement.
| Facility | Mean stay | Median stay | Standard deviation | Range |
|---|---|---|---|---|
| Facility A | 9.2 months | 9.0 months | 1.4 months | 6–12 months |
| Facility B | 9.4 months | 5.0 months | 8.7 months | 1–46 months |
Professional reading.
- The means are nearly identical, so a report citing means alone would conclude the facilities perform the same.
- Facility A is tightly clustered: almost every child moves within a narrow window; the process is consistent.
- Facility B is severely right-skewed: half the children move within five months, but a subgroup is stranded for years, producing a standard deviation larger than its own median.
- The correct managerial question is therefore not "why is B slower?" — it is not slower on average — but "which children are stranded in B, and what characteristic do they share?" That question is generated entirely by reading dispersion rather than central tendency.
The governing lesson for the examination and for practice: never report a mean without a dispersion measure, and never report a skewed distribution without the median.
Two residential facilities report almost identical mean lengths of stay, but Facility A has a standard deviation of 1.4 months while Facility B has a standard deviation of 8.7 months. What should the reviewer conclude?
A student scores 78 on a field rating with mean 70 and standard deviation 8, and 85 on a written test with mean 80 and standard deviation 10. Relative to the group, the student performed:
In a normal distribution, approximately what proportion of cases falls within two standard deviations of the mean?