Consider a normal distribution f(x) of variable x. What percentage of variable values deviate from the mean by more than 2σx​?

Correct option is D

The empirical rule (68-95-99.7 rule) states that for a normal distribution:
· 68.27% of data lies within ±1σ
· 95.45% of data lies within ±2σ
· 99.73% of data lies within ±3σ
Thus, the percentage of values outside ±2σ is:
100%−95.45%=4.55%
Since the normal distribution is symmetrical, the probability in each tail is:

Summing both tails:
2.275%+2.275%=4.55%≈4.56%
Thus, the correct answer is option (d) ~4.56%.
The Z-score formula is:

where X is the data point, μ is the mean, and σ is the standard deviation.
In statistics, outliers are often considered beyond ±2σ or ±3σ.
Bell curve applications: IQ scores, SAT scores, biological measurements, and financial models.
Beyond ±3σ: Less than 0.3% of data falls outside this range.
Additional Information:
· 0.26% (Option a): Represents the probability beyond ±3σ, not ±2σ.
· 2.64% (Option b): Incorrect as it underestimates the probability beyond 2σ.
· 31.74% (Option c): Incorrect as it refers to the probability within a different range.
· 4.56% (Option d): Correct, as it represents the total probability beyond ±2σ.