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Quizzes > Mathematics > Advanced Math

Ready to Ace the Normal Distribution Quiz?

Dive into this quiz on normal distribution and showcase your stats savvy!

Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
Paper art style illustrating normal distribution bell curve quiz elements on sky blue background

Ready to tackle the most iconic curve in statistics? Our free, engaging normal distribution quiz lets you test your grasp of the bell curve, from z-scores to area calculations. Whether you're brushing up on normal distribution questions or diving into a fun quiz on normal distribution, you'll sharpen key skills and build confidence. Ideal for students, educators, and data enthusiasts, this challenge offers targeted prompts, instant feedback, and progress tracking. When you're done, explore our frequency distribution practice problems to keep the momentum going. Get started now and see how well you score!

What is the mean of a standard normal distribution?
0
1
0.5
Undefined
By definition, the standard normal distribution has a mean of zero and a standard deviation of one. This central location sets it apart and makes it the benchmark for z-scores. All other normal distributions can be transformed to this standard form. source
What is the standard deviation of a standard normal distribution?
1
0
2
Undefined
A standard normal distribution is characterized by a standard deviation of one. The standard deviation measures the dispersion of data around the mean. In the standard normal case, this dispersion is fixed, simplifying probability computations. source
Approximately what percentage of values lie within one standard deviation of the mean in a normal distribution?
68%
50%
95%
99.7%
The empirical rule states that about 68% of data in a normal distribution falls within one standard deviation of the mean. This is crucial for quick probability estimates. Two and three standard deviations cover approximately 95% and 99.7%, respectively. source
What shape does a normal distribution curve have?
Bell-shaped
Uniform
J-shaped
U-shaped
The normal distribution is commonly referred to as the bell curve because of its smooth, symmetric bell shape. This shape arises from its probability density function. The peak occurs at the mean, with tails extending to infinity. source
Which of these distributions is symmetric about its mean?
Normal distribution
Exponential distribution
Poisson distribution
Chi-square distribution
The normal distribution is symmetric about its mean, meaning its left and right sides are mirror images. Exponential, Poisson, and chi-square are skewed distributions. Symmetry is a key property for many statistical tests. source
Which symbol typically denotes the standard deviation of a normal distribution?
?
?
?
?
In statistics, ? (sigma) is universally used to denote the standard deviation of a distribution. The symbol ? (mu) denotes the mean. Distinguishing parameters clearly helps avoid confusion. source
If a normal distribution has ? = 50 and ? = 5, what is the z-score for x = 55?
1
0
5
10
The z-score is calculated as (x – ?) / ?. Here, (55 – 50) / 5 = 1. A z-score of 1 indicates that x is one standard deviation above the mean. source
A normal distribution with mean 0 and variance 1 is called:
Standard normal distribution
Gaussian distribution
Uniform distribution
Binomial distribution
A normal distribution with ? = 0 and ?² = 1 is specifically known as the standard normal distribution. 'Gaussian distribution' refers broadly to any normal distribution. The standard form simplifies z-score calculations. source
For Z ~ N(0,1), what is P(Z < 1.96)?
0.9750
0.9500
0.0250
0.5000
The cumulative distribution function (CDF) of the standard normal at 1.96 is approximately 0.975. This value is key for 95% confidence intervals. Tables or software confirm P(Z < 1.96) ? 0.9750. source
What is the formula to convert an observation X to a z-score?
(X - ?) / ?
(X + ?) / ?
(? - X) / ?
? - ?
Z-scores standardize observations by subtracting the mean and dividing by the standard deviation. This transformation yields a standard normal variable. It allows comparison across different normal distributions. source
Given X ~ N(100, 15²), what is P(X > 115)?
0.1587
0.8413
0.5000
0.0228
First compute z = (115 - 100)/15 = 1. Then P(X > 115) = P(Z > 1) ? 1 - 0.8413 = 0.1587. This uses standard normal table values. source
Which z-score corresponds to the upper 5% in a standard normal distribution?
1.645
1.960
2.330
2.576
The z-score that leaves 5% in the upper tail of a standard normal is about 1.645. It is commonly used in one-tailed confidence intervals. 1.96 corresponds to a two-tailed 5% split. source
What is the kurtosis of a normal distribution?
3
0
1
4
Kurtosis measures the 'tailedness' of a distribution. A normal distribution has a kurtosis of 3, often called mesokurtic. Values above or below indicate heavier or lighter tails. source
In the context of the normal distribution, the area under the curve represents:
Probability
Mean value
Variance
Mode
The total area under any probability density function equals 1, representing total probability. For the normal curve, areas under intervals correspond to event probabilities. This is fundamental to statistical inference. source
In a normal distribution, which three measures of central tendency are equal?
Mean, median, and mode
Mean, variance, and skewness
Median, kurtosis, and variance
Mode, range, and skewness
Because the normal distribution is perfectly symmetric, the mean, median, and mode coincide at the center of the distribution. This property simplifies many statistical analyses. In skewed distributions, these measures differ. source
What is the moment generating function M(t) of X ~ N(?, ?²)?
exp(?t + ½?²t²)
exp(?t - ½?²t²)
exp(-?t + ½?²t²)
exp(-?t - ½?²t²)
The moment generating function of a normal random variable is M(t)=E[e^{tX}]=exp(?t + (?²t²)/2). It uniquely characterizes its distribution and is used to find moments. This formula derives via completing the square. source
What is the skewness of any normal distribution?
0
1
3
Undefined
Skewness measures asymmetry. A normal distribution is symmetric about its mean, so its skewness is zero. Deviations from zero indicate a longer tail on one side. source
If X and Y are independent normal random variables, what is the distribution of X + Y?
Normal distribution
Binomial distribution
Uniform distribution
Poisson distribution
The sum of independent normal random variables is also normal. Its mean is the sum of means and its variance is the sum of variances. This closure property is unique to the normal family. source
Given X ~ N(10,4) and Y ~ N(20,9) independent, what is the distribution of X + Y?
N(30,13)
N(30,?13)
N(10,4)
N(20,9)
For independent normals, the sum has mean = 10 + 20 = 30 and variance = 4 + 9 = 13. The notation N(30,13) indicates mean 30 and variance 13. Standard deviation is ?13. source
For Z ~ N(0,1), what is P(-1.96 < Z < 1.96)?
0.9500
0.8413
0.9750
0.6827
Using the standard normal CDF, P(-1.96 < Z < 1.96) ? 0.95. This interval captures 95% of the distribution’s probability. It underpins two-sided 95% confidence intervals. source
At what value of x is the probability density function of N(?, ?²) maximized?
?
? + ?
?
0
The peak of the normal density occurs at its mean ? because the exponential term is zero there. This is the mode of the distribution. It is also where the density is highest. source
What is the value of the normal PDF at its peak when ? = 2?
0.1995
0.3989
0.1599
0.2821
The peak value of a normal PDF is 1/(??2?). With ? = 2, that equals 1/(2?2?) ? 0.1995. For ? = 1, the peak would be 0.3989. Density scales inversely with ?. source
For X ~ N(100, 15²), find the value a such that P(X > a) = 0.025.
129.4
70.6
115.0
140.0
We need a z such that P(Z > z) = 0.025, which gives z ? 1.96. Then a = 100 + 1.96·15 ? 129.4. This finds the 97.5th percentile of X. source
If X ~ N(50, 100) and Y = 3X + 4, what is the distribution of Y?
N(154, 900)
N(154, 100)
N(150, 900)
N(150, 100)
A linear transformation of a normal variable is normal. The new mean is 3·50 + 4 = 154, and the new variance is 3²·100 = 900. So Y ~ N(154, 900). source
For a sample of size n = 36 from a normal population with ? = 12, what is the standard error of the sample mean?
2
12
6
4
The standard error of the mean is ?/?n. Here, 12/?36 = 12/6 = 2. This measures the dispersion of the sampling distribution of the mean. source
0
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Study Outcomes

  1. Understand Normal Distribution Fundamentals -

    Grasp the key properties of the bell curve, including symmetry, mean alignment, and standard deviation in a normal distribution.

  2. Calculate Z-Scores -

    Compute standardized values to determine how many standard deviations a data point lies from the mean.

  3. Interpret Probability Areas -

    Analyze probability regions under the normal curve using standard tables or digital tools to find likelihoods for specific intervals.

  4. Apply Normal Distribution to Real-World Data -

    Use normal distribution principles to solve practical statistical problems and inform data-driven decisions.

  5. Evaluate Distribution Assumptions -

    Assess when it's appropriate to model data with a normal distribution and identify potential deviations or limitations.

Cheat Sheet

  1. Bell Curve Fundamentals -

    The normal distribution is symmetric about its mean, with mean = median = mode, forming the classic "bell curve." According to NIST, approximately 68% of data falls within one standard deviation, 95% within two, and 99.7% within three (the 68-95-99.7 rule). Use the mnemonic "68, 95, 997" to quickly recall these key percentages.

  2. Z-Score Standardization -

    Z-scores transform any normal distribution to the standard normal (mean = 0, σ = 1) using z = (x - μ)/σ. This formula, taught in university courses like those at Khan Academy, lets you compare scores from different scales with ease. Remember: positive z means above the mean, negative means below.

  3. Using the Standard Normal Table -

    A standard normal table (or Z-table) gives cumulative probabilities for z-scores, helping you find P(Z ≤ z). For example, Z = 1.28 corresponds to about 0.8997, so there's an ~89.97% chance a value is below that point. Practice with a "quiz on normal distribution" to become fluent at reverse lookups too.

  4. Empirical Rule vs. Chebyshev's Inequality -

    While the empirical rule (68-95-99.7) applies specifically to normal distributions, Chebyshev's inequality offers broader bounds for any distribution: at least 75% within two SDs and 89% within three. This contrast, highlighted in many academic texts, shows when and why normal distribution questions rely on the empirical rule exclusively. Keep both tools in mind when estimating probabilities.

  5. Practical Applications & Confidence Intervals -

    Normal models underpin confidence intervals: a 95% CI around a sample mean typically is μ̂ ± 1.96·(σ/√n). Industries from finance (risk assessment) to healthcare (clinical trials) rely on these intervals, so mastering these normal distribution questions boosts real-world problem solving. As an extra tip, memorize "1.96" for quick recall of the 95% z-multiplier.

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