Decide whether the normal model is applicable

What should be checked first when deciding whether a histogram supports the normal model?

These ten observations have mean 10 and sample standard deviation about 1.49. Predict the percentage within one standard deviation. Try k = 1, 2 and 3, counting the dots between the boundaries before reading the totals. Displayed boundaries are rounded to two decimal places

What does the observed data show?

Choosing a normal model

The normal model is a useful approximation for a distribution that is approximately symmetric and bell-shaped. Check the whole frequency pattern rather than one bin

1. 1Check approximate symmetry Compare frequencies in bins equally far from the centre. Small differences are acceptable \(\displaystyle 1,3,6,3,2\quad\Rightarrow\quad\text{approximately symmetric}\)
2. 2Check for one central peak A suitable histogram has one main central peak, with frequencies generally falling away on both sides \(\displaystyle 1,3,6,3,1\quad\Rightarrow\quad\text{one central peak}\)
3. 3Check for separated bins A non-zero bin separated from the main group by empty bins can make the normal model unsuitable \(\displaystyle 1,3,5,3,1,0,0,1\quad\Rightarrow\quad\text{isolated high-value bin}\)
4. 4Treat the model as an approximation A roughly symmetric, single bell-shaped peak supports considering a normal approximation. Inspect the whole display, sample size, binning and measurement context. A small histogram cannot prove a population is normal \(\displaystyle \text{rough bell shape}\Rightarrow\text{normal model may be suitable}\)

Given values

\(\displaystyle f_L=1,\quad f_R=1\)

For the outer pair of frequencies 1 and 1, find the absolute difference between the two frequencies

Given values

\(\displaystyle f=1,3,6,3,1\)

A histogram has frequencies 1, 3, 6, 3 and 1. Find the total number of observations

Using the peak alone. One of these lines is right. Which one?

Observed percentages and model estimates

For a supplied data set, calculate the endpoints and count observations inside them. If only an approximately symmetric, bell-shaped distribution is described, the empirical rule estimates the share inside each central interval

1. 1Observed percentage Divide the number inside by the total number of observations, then multiply by 100.
2. 2A central interval The endpoints are equally far below and above the mean: mean minus k standard deviations and mean plus k standard deviations
3. 3Check the shape Symmetry alone is not enough. Do not apply the rule to a strongly skewed, flat or two-peaked distribution

A distribution is symmetric but has two separated peaks. Can symmetry alone justify using the 68–95–99.7% rule?

The histogram shows fictional process A. Which interpretation is supported?

Process B has this symmetric histogram. Is symmetry sufficient for an empirical-rule estimate?

Process C has two separated peaks. What should happen before using one normal model?

Justify a model from complete displays

Four fictional processes record times in consecutive equal-width 10-second intervals starting at zero. Their ordered frequencies are A: 1,3,5,7,5,3,2; B: 4,4,4,4,4,4,4; C: 1,4,7,2,7,4,1; D: 1,3,5,7,5,3,1,0,0,1. Construct all four histograms on common 0–100-second and 0–8-frequency axes. Include lower endpoints and exclude upper endpoints except the final upper endpoint

1. 1Inspect and decide Find each sample size. Describe paired frequencies, the central region and any isolated group. Identify any reasonable normal-model candidate and justify why the other shapes do not support the approximation
2. 2Choose and verify technology Choose a plotting tool and explain the choice. Record numerical boundaries, touching equal-width columns and axis labels. Check every frequency against the supplied table and verify totals. Do not claim to have raw observations; these tables do not determine unique raw records or exact means
3. 3Evaluate limitations Explain why a finite histogram cannot prove normality, why bin width can affect the apparent shape, and why a normal time model can only be an approximation: an exact normal model permits negative values. Explain how you would investigate process D without silently deleting its isolated observation

Model response: justify and qualify the approximation

The sample sizes are A=26, B=28, C=26 and D=26. Plot all supplied frequencies, retaining D’s two empty classes and its final observation. Label the horizontal axis Time (seconds) and the vertical axis Frequency

1. 1Decisions with evidence A is a candidate: paired frequencies are (1,2),(3,3),(5,5), with central frequency 7 and no separated group. B is flat. C has peaks of 7 separated by the central frequency 2. D has a main bell-shaped group and one observation separated by two empty classes. B, C and D do not support a single bell-shaped normal approximation from these displays
2. 2One reproducible workflow Enter lower and upper numerical boundaries with the given frequency columns. Use a frequency-histogram tool accepting grouped counts, or draw touching equal-width columns on the specified numeric scale. Pad absent 70–100-second classes with zeros for A–C on the common window. Check totals with SUM and independently add the seven or ten original counts. A categorical bar chart with gaps is not the requested histogram
3. 3Limits of the conclusion The normal model is continuous and extends without bound in both directions, whereas times cannot be negative. It may be useful over a relevant range when impossible-tail probability is negligible; these tables alone do not quantify that error. Investigate recording, context, mixture and binning before modelling, and retain D’s observation unless evidence justifies a correction. Rebinning exactly requires raw records or compatible known group boundaries

A plot changes appearance when a different bin width is chosen. What is justified?