Game 16 · Describing data and agreement
What does this data look like?
Normal distribution, mean, median, SD and SE, skew, and choosing a graph.
Read with: Descriptive statistics, Data presentation32 DNB questions
Play
Look at the histogram first
Each round shows a made-up dataset. Place the mean and the median on the histogram, set the SD and see the 68-95-99.7 bands, work out the SE and watch it shrink as n grows, and choose the right summary and the right graph. You have three lives.
Loading the first round…
Mean and median
The tail pulls the mean, not the median
The mean is the sum of the values divided by their number. The median is the middle value when the values are put in order. The mode is the most frequent value. In a symmetrical distribution all three are the same.
When a histogram has a long tail on one side, a few extreme values add a lot to the total and pull the mean towards the tail. The median depends only on the order of the values, so it moves very little. In a right-skewed (positive skew) distribution such as hospital stay or income, mode < median < mean. In a left-skewed (negative skew) distribution such as gestational age at delivery, mean < median < mode.
A distribution with two peaks is bimodal. It often means two different groups have been mixed, such as the heights of men and women together. Describe the groups separately, because a single mean or median falls between the two peaks and describes neither group.
SD and the normal curve
68, 95 and 99.7
SD = √[ Σ(x − x̄)² ÷ (n − 1) ]
- x
- each value
- x̄
- the sample mean
- n − 1
- used in a sample, because deviations from the sample mean are slightly smaller than deviations from the population mean
The normal distribution is symmetrical and bell-shaped, and it is fixed completely by its mean and its SD. About 68.3% of values lie within 1 SD of the mean, 95.4% within 2 SD and 99.7% within 3 SD. Exactly 95% lie within 1.96 SD. For systolic pressure with mean 120 and SD 10 mmHg, that gives 110 to 130 (68%), 100 to 140 (95%) and 90 to 150 (99.7%). Mean ± 1.96 SD is used as a reference range only when the data are close to normal in shape.
| Feature | Symmetrical data | Skewed data |
|---|---|---|
| Centre | Mean | Median |
| Spread | SD | Interquartile range (25th to 75th percentile) |
| How to write it | Mean (SD), for example 12.0 (1.41) g/dL | Median (IQR), for example 5 (3–8) days |
The SAMPL guidelines (Lang and Altman, 2013) ask for the form mean (SD), not mean ± SD. See the descriptive statistics chapter.
SD or SE
One describes patients, the other describes the mean
SE = SD ÷ √n
| Feature | Standard deviation (SD) | Standard error of the mean (SE) |
|---|---|---|
| Describes | Variation between individual patients | Precision of the sample mean as an estimate of the population mean |
| As n increases | Stays about the same | Becomes smaller |
| Type of statistic | Descriptive | Inferential (the basis of confidence intervals and t-tests) |
| Use it for | Table 1, baseline characteristics, normal ranges | Confidence intervals. A 95% CI is clearer than the SE itself |
With a population SD of 15 mmHg, means of samples of 25 patients have SE = 15 ÷ √25 = 3 mmHg. Four times the patients halves the SE (15 ÷ √100 = 1.5), but the SD of the patients stays about 15.
Myth: The SE looks smaller and tidier, so it is better after the ± sign.
The SE describes the precision of the mean, not how patients vary. Use the SD to describe patients.
Myth: A larger sample has a smaller SD.
A larger sample estimates the SD more precisely, but the SD stays about the same. Only the SE shrinks.
Choosing a graph
The type of data decides the graph
| Data | Graph |
|---|---|
| Shape of one continuous variable (birth weight) | Histogram: class intervals on a continuous scale, bars touching |
| Categories (blood group, anaemia grade) | Bar chart: one bar per category, gaps between bars, axis from zero |
| Parts of one whole, few categories | Pie chart: flat, a few slices only |
| Skewed values compared between groups | Box plot: median, quartiles, whiskers and outliers on one scale |
| Two measurements on the same people | Scatter plot: one dot per person |
| Change over time | Line graph: time on the horizontal axis, points joined in order |
A histogram and a bar chart look alike but are not the same. A bar chart shows separate categories, so its bars have gaps. A histogram shows a continuous scale, so its bars touch and the area of each bar is proportional to its frequency. The data presentation chapter covers each graph.
Writing the answer
How to write this in a 10-mark DNB answer
- For the normal distribution, draw the bell curve first, with the mean in the middle and ±1, ±2 and ±3 SD marked with 68%, 95% and 99.7%.
- List its properties: symmetrical, mean = median = mode, fixed by the mean and SD, tails never reach zero. Give the z-score: z = (x − mean) ÷ SD.
- Draw the two skewed shapes beside it, with the order of mean, median and mode marked on each.
- For measures of dispersion, define range, IQR, variance, SD and CV with their formulas, and work one small example.
- For SD versus SE, give SE = SD ÷ √n and a comparison table (describes, changes with n, use), with one example.
- End with the reporting rule: mean (SD) for symmetrical data, median (IQR) for skewed data, and never the SE to describe patients.
The one thing to remember
Draw the histogram before you choose a summary. Its shape decides between mean (SD) and median (IQR).
Past papers
DNB questions this game helps you answer
Questions from NBE theory papers for DNB, DrNB and Diploma courses, most often asked first. The tag after each question is the specialty and the year it was set.
- ×2Standard deviation.
- ×2Standard deviation
- ×1What is a normal distribution curve?
- ×1What is normal distribution of data? What are the types of variability in data?
- ×1What is the difference between standard deviation and standard error?
- ×1Box plot (box and whisker plot).
- ×1Measure of central tendency and dispersion in data analysis.
- ×1a) Confidence interval. b) Tests to evaluate normal distribution of data. c) Sample size calculation.
- ×1a) Measures of Central Tendency. b) Parametric and non parametric tests. c) Analysis of variance.
- ×1Gaussian distribution.
- ×1Interpretation of Gaussian curve.
- ×1Measures of dispersion in statistics.
- ×1What is normal distribution? What are the prerequisites for applying parametric tests?
- ×1What is normal distribution? Describe the tests of significance which can be applied on normally distributed data.
- ×1Define Standard error. Give any 4 examples to illustrate its application in test of significance.
- ×1Standard deviation in statistical data.
- ×1What are the measures of dispersion? Describe each one with examples.
- ×1What is the Standard normal curve (or normal distribution)? Discuss the measures of dispersion in data.
- ×1Define the following and give their formulae: b) The standard deviation.
- ×1What is Standard Deviation (SD)?
- ×1a) One-way ANOVA and its Post-hoc tests. b) Tests to check normal distribution of data.
- ×1Mean, median, mode and standard deviation.
- ×1Concept of Standard Normal Distribution and its relation to the various tests of significance.
- ×1Gaussian Distribution and Poisson Distribution
- ×1Measures of Dispersion
- ×1Histogram.
- ×1Differentiate between standard deviation and standard error in case of quantitive data.
- ×1Use of mean, median and mode in descriptive statistics.
- ×1What is normal distribution?
- ×1Application of “Measures of Dispersion” in Biostatistics.
- ×1Measures of central tendency in a series of observations; Measures of variability of individual observations.
- ×1Standard Error
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