DAT Quantitative Reasoning: The Normal Distribution and the 68-95-99.7 Rule

DAT High-Yield Concept

The 68-95-99.7 Rule (Empirical Rule): In a symmetric, bell-shaped normal distribution, approximately 68% of data points fall within 1 standard deviation of the mean, 95% fall within 2 standard deviations, and 99.7% fall within 3 standard deviations.
Symmetry Halving: Splitting symmetric bands around the mean ( $\mu$ ) yields one-sided percentages, such as exactly 34% lying between $\mu$ and $\mu + 1\sigma$ .
Tail Areas: Tail probabilities are determined by subtracting central bands from 100% and dividing by 2. For instance, exactly 2.5% of values lie more than 2 standard deviations above the mean ( $\mu + 2\sigma$ ).
Sample Count Conversion: Convert a distribution percentage into an actual subject count by multiplying the percentage (in decimal form) by the sample size ( $N$ ).

1. Anatomy of the Normal Curve and the Empirical Rule

The normal distribution is a continuous probability distribution defined by a bell-shaped density curve that is perfectly symmetric about its mean ( $\mu$ ). The width and spread of the curve are dictated by the standard deviation ( $\sigma$ ). On the DAT Quantitative Reasoning section, standard normal calculations rely on the 68-95-99.7 Rule (also known as the Empirical Rule):

\text{Interval } [\mu - 1\sigma, \mu + 1\sigma] \approx 68\% \text{ of total area}

\text{Interval } [\mu - 2\sigma, \mu + 2\sigma] \approx 95\% \text{ of total area}

\text{Interval } [\mu - 3\sigma, \mu + 3\sigma] \approx 99.7\% \text{ of total area}

The standard score or z-score represents the distance of a value from the mean in standard deviation units ( $z = \frac{x - \mu}{\sigma}$ , covered in depth in its own lesson). Consequently, the empirical rule boundaries correspond directly to integer z-scores of $z = \pm 1$ , $z = \pm 2$ , and $z = \pm 3$ .

68, 95, 99.7 -- and the eight slices they are built from — A normal curve is symmetric about its mean, and the empirical rule fixes the areas: about 68 percent of values lie within one standard deviation of the mean, about 95 percent within two, and about 99.7 percent within three. Symmetry halves any band, so about 34 percent lies between the mean and one standard deviation above it. A tail is the leftover halved, so about 2.5 percent lies more than two standard deviations above the mean. Multiplying a percent by the sample size turns it into a count: with a mean of 500, a standard deviation of 100 and 2000 test-takers, about 1360 score between 400 and 600 and about 50 score above 700.

A visual representation of the Empirical (68-95-99.7) Rule based on the normal distribution. (Source: Dan Kernler · CC BY-SA 4.0 · Wikimedia Commons)

Key Concept: Bell Curve Symmetry
Because the normal distribution is symmetric about

\mu

, the area to the left of the mean is exactly 50% (0.50) and the area to the right is exactly 50% (0.50). Every central interval

\mu \pm k\sigma

is divided into two equal left and right halves.

2. Halving Bands and Segmenting Slices

By applying symmetry, we can divide the standard normal curve into eight distinct slices bounded by standard deviation increments:

Between $\mu$ and $\mu + 1\sigma$ : $\frac{68\%}{2} = 34\%$
Between $\mu + 1\sigma$ and $\mu + 2\sigma$ : $\frac{95\% - 68\%}{2} = \frac{27\%}{2} = 13.5\%$
Between $\mu + 2\sigma$ and $\mu + 3\sigma$ : $\frac{99.7\% - 95\%}{2} = \frac{4.7\%}{2} = 2.35\%$
Beyond $\mu + 3\sigma$ (Outer Tail): $\frac{100\% - 99.7\%}{2} = \frac{0.3\%}{2} = 0.15\%$

Empirical Rule for Normal Distribution simplified version based on 68, 95, 99.7% values (Source: Holschrk · CC BY-SA 4.0 · Wikimedia Commons)

Standard Deviation Band	Central Percentage	One-Sided Slice Percentage	Cumulative Area Above $\mu$
Within $1\sigma$ ( $\mu \pm 1\sigma$ )	68%	34%	84% (50% + 34%)
Within $2\sigma$ ( $\mu \pm 2\sigma$ )	95%	13.5% (from $1\sigma$ to $2\sigma$ )	97.5% (50% + 47.5%)
Within $3\sigma$ ( $\mu \pm 3\sigma$ )	99.7%	2.35% (from $2\sigma$ to $3\sigma$ )	99.85% (50% + 49.85%)

DAT Memory Tip: 34 - 13.5 - 2.35 - 0.15
Memorize these four single-sided slice percentages moving outward from the mean. Any interval or tail question can be solved rapidly by adding or subtracting these base slices!

3. Upper and Lower Tail Calculations

Tail areas represent values falling beyond a specified standard deviation cutoff. To calculate a upper or lower tail percentage, subtract the central band from 100% and divide by 2:

\text{Tail Percentage} = \frac{100\% - \text{Central Percentage}}{2}

For example, the percentage of values lying more than 2 standard deviations above the mean ( $x > \mu + 2\sigma$ ) is $\frac{100\% - 95\%}{2} = 2.5\%$ (consisting of $2.35\% + 0.15\%$ ).

Worked Example 1: Finding Tail Probabilities

Problem: A study measures resting heart rates in pre-dental students and finds a normal distribution with a mean of 70 bpm and a standard deviation of 8 bpm. What percentage of students have a heart rate greater than 86 bpm?

Step-by-Step Solution:

Identify the mean and standard deviation: $\mu = 70$ bpm, $\sigma = 8$ bpm.
Determine the cutoff position: $86\text{ bpm} = 70 + 2(8) = \mu + 2\sigma$ .
Identify the target region: values greater than $\mu + 2\sigma$ (the upper tail).
Apply the tail formula: $\frac{100\% - 95\%}{2} = 2.5\%$ .

Final Answer: 2.5% of students have a heart rate greater than 86 bpm.

Common Exam Trap: Students often forget to divide the remaining 5% by 2 and select 5% as their answer. Remember that 5% represents both tails combined ( $< 54\text{ bpm}$ and $> 86\text{ bpm}$ ).

4. Converting Percentages to Counts in Sample Sizes

DAT questions frequently require converting probability percentages into concrete subject counts for a sample of size $N$ :

\text{Predicted Count} = \left( \frac{\text{Percentage}}{100} \right) \times N

Worked Example 2: Sample Count Calculation

Problem: An entry exam with $N = 500$ applicants produces normally distributed scores with a mean of 150 and a standard deviation of 15. How many applicants scored between 135 and 180?

Step-by-Step Solution:

Find standard deviation positions for the bounds:
- Lower bound: $135 = 150 - 15 = \mu - 1\sigma$
- Upper bound: $180 = 150 + 2(15) = \mu + 2\sigma$
Sum the corresponding slices between $\mu - 1\sigma$ $μ - 1 σ$ and $\mu + 2\sigma$ $μ + 2 σ$ :
- Slice $[\mu - 1\sigma, \mu]$ : 34%
- Slice $[\mu, \mu + 1\sigma]$ : 34%
- Slice $[\mu + 1\sigma, \mu + 2\sigma]$ : 13.5%
- Total percentage = $34\% + 34\% + 13.5\% = 81.5\%$
Convert percentage to count for $N = 500$ : $\text{Count} = 0.815 \times 500 = 407.5 \approx 407.5 \text{ applicants}$

Final Answer: Exactly 407.5 (or approximately 407 to 408) applicants.

Common Exam Trap: Selecting 68% or 95% directly without checking whether the upper and lower bounds are symmetrical around the mean. Always evaluate each boundary individually!

Critical Thinking Questions

If a dataset of 1,000 scores has 160 values greater than $\mu + 1\sigma$ , does this dataset follow a normal distribution? Explain using tail percentage rules.
How does changing the standard deviation $\sigma$ affect the total area under the curve and the percentage of values contained within $\mu \pm 1\sigma$ ?
A patient's blood test value lies at the 84th percentile. What is the corresponding standard deviation distance (z-score) from the mean in a normal distribution?

References

Sources returned by Google Search grounding during lesson verification.

68–95–99.7 rule - Wikipedia — en.wikipedia.org
11.9: The Empirical Rule - Mathematics LibreTexts — math.libretexts.org

The Normal Distribution and the 68-95-99.7 Rule

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