Use z-score boundaries to estimate normal-model percentages and counts

An actual boundary value is given with a mean and standard deviation. What is the first calculation?

This is a bell-shaped model. Keep k = 1 and change s from 2 to 10. Then try k = 2 and 3. Compare the shaded area and endpoints. Percentages describe area, not the height of the curve

What changes the estimated percentage in the central shaded interval?

Estimate normal-model percentages

A normal-model percentage question may give z-score boundaries or actual boundary values. Convert actual values to signed z-scores, identify the requested region, and then use the empirical rule; scale the percentage by the population size only for an expected count

1. 1Standardise an actual boundary Subtract the mean from the actual value, then divide by the standard deviation \(\displaystyle z=\frac{x-\mu}{s}\)
2. 2Keep the sign A value below the mean has a negative z-score, while a value above the mean has a positive z-score \(\displaystyle x<\mu\Rightarrow z<0,\quad x>\mu\Rightarrow z>0\)
3. 3Match the region Use the signed boundaries to decide whether the required values are in a central interval, one tail or two tails \(\displaystyle P(-2\le z\le2)\approx95\%,\quad P(z\ge2)\approx2.5\%\)
4. 4Scale for a count Multiply the estimated percentage as a proportion by the population size only when an expected count is required \(\displaystyle \frac{95}{100}\times4000=3800\)

Given values

\(\displaystyle x=50,\quad\mu=60,\quad s=5\)

A distribution has mean 60 and standard deviation 5. Find the z-score for the boundary value 50.

Given values

\(\displaystyle -2\le z\le2,\quad\text{normal model; empirical rule}\)

Assume an approximately normal model. Using the 68–95–99.7% rule, estimate the percentage between z=−2 and z=2.

For boundary x=50, mean 60 and standard deviation 5, which numerator order gives the signed standard score?

The 68–95–99.7% rule

Use these approximate central percentages only for an approximately symmetric, bell-shaped distribution. The endpoints are mean ± k standard deviations

1. 1One standard deviation About 68% lies between mean − s and mean + s.
2. 2Two standard deviations About 95% lies between mean − 2s and mean + 2s
3. 3Three standard deviations About 99.7% lies between mean − 3s and mean + 3s

Central percentage

\(\displaystyle 95\%\)

Approximately 95% lies in a central interval. What percentage lies outside it, counting both tails?

Identify the interval or side being asked about before calculating

Both tails combined

\(\displaystyle 5\%\)

A symmetric bell-shaped distribution has 5% outside its central 95% interval. What percentage is in just one tail?

Identify the interval or side being asked about before calculating

A student answers 2.5% for the percentage below mean + 2s. What went wrong?

Does the empirical rule guarantee exactly 95 of every 100 observations are within two standard deviations?

Map boundaries before combining areas

Sketch a labelled normal-model axis, mark the mean, convert every actual boundary using the standard-score formula, and shade the requested region. Use the stated 68–95–99.7% approximation at z=±1,±2,±3. Each tail is half of the percentage outside a central band

1. 1Mixed and outside regions For the interval from two standard deviations below the mean to one above it, combine 47.5% below the mean with 34% above it. The outside region is its complement. For one upper tail beyond two standard deviations above the mean, halve the 5% outside the central 95% rather than using the central percentage
2. 2Respect the model and method These are approximate areas, not curve heights or guaranteed observed fractions. A continuous normal model gives zero probability to an exact endpoint, so strict versus inclusive endpoints do not change its area; actual discrete recorded counts can differ. The rule does not supply an exact percentage at z=1.5. Use a normal CDF only when that method is requested, documenting its different approximation

Given values

\(\displaystyle P(-2\le z\le0)\approx47.5\%,\quad P(0\le z\le1)\approx34\%\)

Use the empirical rule for an approximately normal model. What percentage lies between z=−2 and z=1?

Given values

\(\displaystyle p_{\mathrm{inside}}=81.5\%\)

For that model, approximately 81.5% lies between the two boundaries. What percentage lies outside them?

Given values

\(\displaystyle n=50,\quad p=0.815\)

A model assigns probability 0.815 to an interval. For 50 future observations, give the unrounded expected count

Convert, shade, estimate and verify

Assume a normal model for fictional process durations with mean 100 seconds and standard deviation 10 seconds. Use the 68–95–99.7% rule throughout. Sketch and label the actual and z boundaries for 80–110 seconds, outside that interval, above 120 seconds and below 70 seconds. Find each approximate percentage and the expected count among 200 observations

1. 1Reverse and round Find the threshold exceeded by about 2.5%. For a separate group of 50, find the unrounded expected count between 80 and 110 seconds and then report a nearest-whole-number estimate. Explain why no count is guaranteed
2. 2Choose and verify technology Choose calculation or spreadsheet functionality and document the mean, spread, signed boundaries, region, percentage convention and count formula. Verify by adding disjoint empirical-rule segments and checking complementary percentages sum to 100. Explain why interpolation to z=1.5 would not be an exact rule result
3. 3Complete before continuing Complete writing, drawing and external-tool execution are not automatically marked; use self or teacher review

Model response: distinguish areas and counts

The 80–110-second interval maps to −2≤z≤1. Its approximate area is 47.5+34=81.5%, giving expected count 163 of 200. The outside region has area 18.5%, giving 37. Above 120 seconds is z>2: 2.5%, giving 5. Below 70 seconds is z<−3: 0.15%, giving expected count 0.3

1. 1Sketch and reverse Draw one bell-shaped curve with mean 100, standard-deviation ticks 70,80,90,100,110,120,130 and matching z ticks −3,−2,−1,0,1,2,3. Use separate shading for each requested region. The threshold with about 2.5% above it is 100+2×10=120 seconds. Percentages describe shaded areas, not the curve’s height
2. 2Counts and precision For 50 observations, 50×0.815=40.75, or about 41 to the nearest whole observation. The fractional expectations 40.75 and 0.3 are legitimate averages under the model; they are not possible literal observed counts or guarantees. Report approximate percentages and distinguish expectations from observed records
3. 3One reproducible workflow Enter actual boundaries in labelled cells and calculate =(boundary-100)/10. Store empirical cumulative percentages at z=−3,−2,−1,0,1,2,3 as 0.15,2.5,16,50,84,97.5,99.85. Subtract the two cumulative percentages for an interval, complement for outside and multiply by n/100 for expected counts. Verify 81.5+18.5=100 and 2×0.15+99.7=100. A numerical normal CDF is a different calculation; the rule alone does not give the z=1.5 area

The empirical rule gives an expected count of 0.3 below a boundary in 200 observations. What does that mean?