The Normal Model and the 68-95-99.7 Rule
Two slides. The first states three percentages and a sketching routine; the second is a picture with nothing recoverable on it. This page reproduces the first exactly, and then says plainly why you cannot act on it from this material alone.
The rule, exactly as the source states it
THE RULE AS THE SOURCE STATES IT
| Distance from the mean | Proportion of all observations stated to fall inside |
|---|---|
| One standard deviation | 68% |
| Two standard deviations | 95% |
| Three standard deviations | 99.7% |
The three percentages are quoted by the slide from a published business research methods text of 2005, at p. 462 — the only figures in the week attributed to an identified work with a page number. Everything else on the slide is the deck's own.
That is the whole of the rule. It is the one quantitative statement in a week devoted to quantitative analysis, and it is the only place in the material where a numeric claim about data is attributed to a published source rather than asserted.
What the slide does with the rule: a sketching procedure
The use the slide puts the rule to is not interpretation. It is drawing. The rule is presented as a way of getting the axis of a hand-sketched bell curve to scale, in three steps.
The sketching procedure, as stated
Draw the bell shaped curve
The symmetrical smooth curve the previous slide describes as the basis for the normal probability model.
Mark the centre with the value of the mean
The mean is defined earlier in the week, in words, with no formula and no worked calculation — see Measures of Central Tendency.
Mark off three standard deviations either side
Three either side is a drawing requirement stated by the slide. The distance one standard deviation represents is what sets the scale on the axis.
Two numbers appear in that procedure and they are different kinds of thing. Three standard deviations either side is a requirement of the drawing instruction. The 5 units is a worked-example value, supplied by the slide for one illustration — it is the only numeric quantity in the deck's entire treatment of the normal distribution, and it is not a standard of any kind.
The rule is written in a unit the week never defines
Every line of the rule is a statement about a distance measured in standard deviations. The term is used four times on this one slide. It is defined nowhere in the week, and the places a reader would reasonably look are all places where it could have been and is not.
WHERE A READER WOULD LOOK FOR THE MISSING DEFINITION
| Where you would look | What is actually there |
|---|---|
| The week's key terms slide | Five terms, all concerned with data management, screening, coding and outliers. The standard deviation is not among them |
| The terminology table for quantitative data analysis | Eight terms: two branches of statistics, central tendency and its three measures, a count and a display. No measure of dispersion of any kind appears in it |
| The slide naming the key features of a distribution | Declares spread one of three key features and calls twice for "some measure of spread". Names none, and does not mention the standard deviation |
| Anywhere else in the deck | No formula, no symbol, no notation, no computed value, no interpretation and no worked calculation for it. The week's only formula is the histogram class rule |
What a second source has to supply before you can use this
Take this list to a statistics or research methods text and satisfy every line of it. Each item is something the rule silently assumes and this material does not provide.
The gap, itemised as a shopping list
- A definition of the standard deviation — what quantity it is, and what it is a measure of
- How it is computed from a set of observations, and whether the form you are using is the one intended for a sample or for a whole population
- What units it is expressed in, so that a distance of one or two of them means something in the units you actually measured
- The condition under which the three percentages hold, stated explicitly — the slide states them unconditionally and its own title supplies the only context
- A way of checking whether your own data meet that condition, since the week contains no test of any kind
- What to do, and what to report, when the condition is not met
- The distinction between a parameter and a statistic — the week defines parameter and never introduces statistic, so the pairing on which its own account of inference depends is missing; see Descriptive and Inferential Statistics
The condition the slide does not print
Read the three lines again as sentences. "68% of all observations will fall within one standard deviation of the mean" is unconditional and exact. It says will, not may; it says all observations, without saying of what; and it carries no qualifier at all.
The slide is headed as a normal distribution, so the context is supplied by the title. The statements themselves are not qualified, the word approximately does not appear, and the week never states what makes a distribution normal in the first place — the phrase normal probability model is named on the previous slide and defined nowhere, as recorded at Reading a Graphical Display.
Where the rule is not used, and could have been
The rule is offered as a drawing aid — "We can use this rule to sketch a normal distribution" — and it is never once used to answer a question about data. No value is converted into a distance in standard deviations; no proportion is read off; the words probability, percentile and z-score appear nowhere in the week.
The clearest missed application is the deck's own. Three earlier slides deal with extreme values and ask you to decide whether to keep or drop them, and they give no detection rule of any kind — the only test available is whether a value looks implausible. A rule expressing distance from the mean in standard deviations is the natural basis for such a test, and the deck holds both halves and never puts them together. See Handling Outliers for the three positions the week takes on extreme values.
Using the week's normal-distribution material honestly
The source does not prescribe the following; it is how to write about this material without claiming more than it contains.
What to do with each situation the rule creates
What to carry forward
- The rule as stated: 68% within one standard deviation of the mean, 95% within two, 99.7% within three — quoted by the slide from a 2005 business research methods text at p. 462.
- It is the week's only quantitative rule and it is written entirely in a statistic the week never defines. Four uses on one slide, no definition anywhere.
- You cannot apply it from this material. Get a definition, a computation, the units, the condition under which the percentages hold and a way of checking that condition — from a text you cite.
- The percentages are stated as unconditional facts about all observations, with no qualifier, in a week that has just taught you to look for skew, multiple peaks and clusters.
- The rule is used only to draw a curve. It is never used to answer a question about data, and never applied to the extreme-value problem it would have fitted.
Frequently asked questions
What does the 68-95-99.7 rule say in this material?
That 68% of all observations will fall within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three. The slide attributes those figures to a 2005 business research methods text at p. 462, and uses the rule to set the scale when sketching a bell-shaped curve.
Can I use the rule on my own data after reading this material?
No. Every line of it is a distance measured in standard deviations, and the standard deviation is used four times on that slide and defined nowhere in the week. Without a definition and a way of computing it, there is no quantity for the rule to be about.
Why does this page not just give the formula?
Because the formula is not in the supplied material, and printing it here would imply the material contains it. The library's rule on this batch is to name a gap accurately rather than fill it from general knowledge, so that a reader knows they need a second source instead of believing they already have one.
What exactly do I need to get from a second source?
A definition of the standard deviation; how it is computed and whether the sample or population form applies; the units it is expressed in; the condition under which the three percentages hold; a way to check your data against that condition; and what to do when they fail it. The parameter and statistic distinction is worth adding, since the week defines only the first.
Do the percentages hold for any set of data?
The slide states them without any condition attached — "will fall", about "all observations" — and the only context is the slide's own title. The week never states what makes a distribution normal, never uses the word approximately, and never says what to do when data are not symmetrical, so the material itself cannot answer this question for you.
What is the 5 units mentioned on the slide?
A worked-example value for a standard deviation, supplied so that the sketch has a scale. No dataset, variable, mean or unit of measurement is given for it, no calculation produces it, and the diagram on the following slide has no recoverable content. It is an illustration, not a standard, and it cannot be reproduced.
References and source attribution
- Veal, A. J. 2005, Business Research Methods: A Managerial Approach, 2nd ed., Longman, p. 462. — the source the slide quotes for the three percentages, and the sole entry on the week's reference slide.
- A publisher's companion website for a fourth-edition social research text, cited on the week's terminology slide as a bare address with no author, year or title stated. It supplies the week's definitions of the mean, median and mode, and no measure of dispersion.
- The prescribed text, Chapter 16, pp. 353–374 — the week's only set reading, on statistical software. It is not a reading on quantitative analysis, and no author, year, title or publisher is given for it in the week's slides.
- The supplied teaching source: the week 11 slide deck on analysing data and the presentation of quantitative data, whose normal-distribution slide is reproduced in full on this page. No study notes for this week were supplied, and the accompanying diagram slide carries no recoverable content.
Suggested questions for Ask KEVOS
- What does the supplied material actually state about the normal distribution?
- List everything I need from a statistics text before I can apply the 68-95-99.7 rule.
- Help me word a methodology sentence that cites this rule without over-claiming.
- Why can't I use this rule to detect outliers in my dataset from this material?
- What should I say in my limitations section about the measures this material does not supply?