Mean, median, mode, standard deviation, normal distribution
High School Essentials • 9-12
Statistics transforms raw data into meaningful insights. In this lesson we review measures of center, introduce the concept of spread through standard deviation, and explore the most important distribution in all of statistics: the normal distribution.
| Measure | Definition | Best Used When |
|---|---|---|
| Mean | Sum of all values divided by the count | Data is symmetric, no extreme outliers |
| Median | Middle value when data is ordered | Data is skewed or has outliers |
| Mode | Most frequently occurring value | Categorical data or identifying peaks |
The standard deviation (denoted σ for a population, s for a sample) measures how spread out data values are from the mean. A small standard deviation means data clusters tightly around the mean; a large one means data is widely scattered.
To calculate standard deviation:
Find the population standard deviation of: 4, 8, 6, 5, 7.
The normal distribution (bell curve) is symmetric, centered at the mean μ, with spread determined by σ. It arises naturally when many small, independent random effects combine -- heights, test scores, measurement errors, and countless other phenomena follow approximate normal distributions.
For normally distributed data:
This means only 5% of data lies more than 2 standard deviations from the mean, and values beyond 3σ are extremely rare (0.3%).
Test scores are normally distributed with mean 72 and standard deviation 8. What range contains 95% of scores?
A z-score tells you how many standard deviations a value is from the mean:
A z-score of 0 means the value equals the mean. Positive z-scores are above the mean; negative z-scores are below.
In a class with mean score 80 and standard deviation 5, a student scores 92. What is her z-score?
Z-scores let you compare values from different distributions. A z-score of 1.5 on a math test and a z-score of 2.0 on a reading test tell you the student performed relatively better on the reading test, even though the raw scores and scales are completely different.
Applying the empirical rule to data that is not approximately normal. If the data is heavily skewed, bimodal, or has a very different shape, the 68-95-99.7 percentages will not apply. Always check that the distribution is roughly bell-shaped first.
Mean = 45/7 ≈ 6.43. Ordered: 3, 5, 5, 5, 7, 9, 11. Median = 5. Mode = 5.
z = (42 - 50)/4 = -8/4 = -2. The value is 2 standard deviations below the mean.
64 = 70 - 2(3) and 76 = 70 + 2(3). This is within 2 standard deviations, so approximately 95%.
First: z = (88 - 75)/5 = 2.6. Second: z = (92 - 82)/4 = 2.5. The first score (z = 2.6) is slightly more unusual.
68% is within ±1σ, so 32% is outside. By symmetry, 16% is above μ + σ.