Normal Distribution
The normal distribution, also known as the Gaussian distribution, is characterised by its bell-shaped curve and relevance towards applied mathematics. It is important as it is the outcome of a probabilistic event with finite mean and variance provided that it is repeated infinitely.
Properties of the Normal Distribution
- Symmetry: The normal distribution is perfectly symmetrical around its mean (μ).
- Mean, Median, and Mode: In a normal distribution, the mean, median, and mode are all equal.
- Continuous Distribution: The normal distribution is a continuous probability distribution.
- Asymptotic Behavior: The tails of the distribution extend infinitely in both directions, approaching but never touching the x-axis.
The normal distribution is often referred to as the "bell curve" due to its characteristic shape.
Graph Representation
The normal distribution curve is as a smooth, symmetrical bell-shaped curve. The highest point of the curve corresponds to the mean (μ), which is also the median and mode of the distribution. The spread of the curve is determined by the standard deviation (σ).
TipWhen sketching a normal distribution curve, remember that it should be symmetrical and that the tails should never completely touch the x-axis.
Natural Occurrence
Normal distributions are ubiquitous in nature and human-related phenomena. Some examples include:
- Heights of individuals in a population
- Blood pressure readings
- IQ scores
- Measurement errors in scientific experiments
- Test scores in large populations
The distribution of adult male heights in a large population often follows a normal distribution. If the mean height is 175 cm with a standard deviation of 7 cm, we can expect most heights to fall within the range of 161 cm to 189 cm (μ ± 2σ).
ExampleLet's consider a normally distributed dataset of exam scores with a mean of 70 and a standard deviation of 5.
- About 68% of students will score between 65 and 75 (70 ± 5)
- About 95% of students will score between 60 and 80 (70 ± 10)
- About 99.7% of students will score between 55 and 85 (70 ± 15)
Students often confuse the 68-95-99.7 rule with exact percentages. Remember, these are approximations and the actual percentages may vary slightly in real-world applications.
Normal Probability Calculations Using Technology
Students are expected to use technology (such as graphing calculators or statistical software) to perform normal probability calculations.
ExampleUsing a graphing calculator (e.g., TI-84), we can calculate the probability that a randomly selected value from a normal distribution with μ = 50 and σ = 10 is less than 60:
- Press [2nd] [DISTR]
- Select "normalcdf("
- Enter: normalcdf(-1E99, 60, 50, 10)
- The result is approximately 0.8413 or 84.13%
This means there's about an 84.13% chance that a randomly selected value will be less than 60.
TipWhen using technology for normal probability calculations, always check that you're entering the parameters in the correct order: lower bound, upper bound, mean, standard deviation.
Inverse Normal Calculations
Inverse normal calculations involve finding the value of the variable that corresponds to a given probability. This is particularly useful in determining percentiles.
ExampleUsing a graphing calculator to find the 90th percentile of a normal distribution with μ = 100 and σ = 15:
- Press [2nd] [DISTR]
- Select "invNorm("
- Enter: invNorm(0.90, 100, 15)
- The result is approximately 119.22
This means that 90% of the values in this distribution are below 119.22.
NoteIn inverse normal calculations, you're given the probability (area) and you're finding the corresponding x-value on the distribution.
