What is a bell curve (normal distribution)?
– A bell curve is the familiar, symmetric “bell-shaped” graph used to describe how values are distributed around a central value. In the idealized case, most observations cluster near the center (the average), and progressively fewer observations appear as you move farther from that center. In statistics this shape is called the normal distribution.
Key definitions (first use)
– Mean: the arithmetic average of all observations; the center of a normal distribution.
– Median and mode: in a perfectly symmetric normal distribution, these equal the mean.
– Standard deviation (σ): a measure of how spread out the data are around the mean. A larger σ produces a wider bell.
– Z‑score: the number of standard deviations a value lies from the mean; z = (x − μ)/σ.
– Volatility: in finance, the standard deviation of returns; it quantifies how much returns vary over time.
– Skewness: a measure of asymmetry in a distribution.
– Kurtosis: a measure of the “tailedness”; excess kurtosis indicates fatter tails than the normal curve.
How the bell curve behaves (practical points)
– Symmetry: the distribution is mirror-symmetric around the mean.
– Empirical rule (useful shortcut): about 68% of observations lie within ±1σ of the mean; about 95% within ±2σ; about 99.7% within ±3σ. These are approximate percentages that apply exactly only when the distribution is normal.
– Probability via z‑scores: converting a value to a z‑score lets you look up probabilities in standard normal tables or compute them with software.
Worked numeric example (simple, step-by-step)
Suppose test scores for a class have a mean μ = 75 and standard deviation σ = 8.
1. Range within 1σ: 75 ± 8 → 67 to 83. Expect roughly 68% of students in this range.
2. Range within 2σ: 75 ± 16 → 59 to 91. Expect roughly 95% in this range.
3. If a student scores 91, compute z = (91 − 75)/8 = 16/8 = 2. A z of +2 means the score is two standard deviations above the mean; the probability of exceeding this (upper tail) is about 2.5% under normality (so roughly 97.5% are at or below 91).
How bell curves