Normal Distribution — Basic

Normal Distribution — Basic. Practice questions to deepen understanding of the normal distribution at the basic level. Online statistics practice with full solutions and step-by-step explanations.

Basic normal distribution practice — 40 questions: bell curve, standard deviation, z-score, tables, the 68-95-99.7 rule. Visual explanations.

Practice the foundations of the normal distribution — understanding the bell curve, standard deviation, the standard score (z).

40 questions

Question 1

2.50 pts

🎯 What is a normal distribution?
Choose the most appropriate definition for a normal distribution.

A normal distribution is a continuous distribution shaped like a "bell", symmetric around the mean, where most values are close to the mean and few values are far from it.

The normal distribution is any table containing percentages whose total is 100%.

The normal distribution is any situation where all values are exactly equal to the mean.

The normal distribution is one in which all values are equally likely (uniform distribution).

Explanation:

💡 Explanation:

Everyday language:
A normal distribution is a "bell curve" – most students fall around the mean, and very few are very high or very low. The graph looks like a symmetric hill in the middle.

Mathematical language:
A normal distribution is a continuous probability distribution, symmetric around the mean μ. The highest density is around μ, and the probability decreases as one moves away from the mean on both sides.

Therefore the correct answer is the one describing a bell curve symmetric around the mean.

Question 2

2.50 pts

📐 Symmetry in the normal distribution:
What does it mean that the normal distribution is symmetric around the mean μ?

The probability of getting a value greater than the mean by some amount equals the probability of getting a value less than the mean by the same amount (the same distance).

The probability of getting a value greater than the mean is always 0.

The probability of getting a value less than the mean is always greater than the probability of getting a value greater than the mean.

The normal distribution is not symmetric at all; one side is always higher.

Explanation:

Everyday explanation:
If a distribution is symmetric, it means that if we look at how many students are 10 points above the mean, there will be roughly the same number of students 10 points below the mean.

Mathematical explanation:
In a normal distribution, symmetry means:
P(X > μ + a) = P(X < μ - a) for every distance a > 0 from the mean.

Therefore the correct answer is that probabilities on both sides at equal distance from the mean are identical.

Question 3

2.50 pts

⚖️ Position of central tendency measures:
In a perfectly normal distribution, what is true about the mean, median, and mode?

They coincide – the mean, median, and mode are all equal to each other.

The mean is always greater than the median, and the median is greater than the mode.

The mode is always greater than the mean, and the median lies between them.

There is no relationship between the three measures – each is in a different place.

Explanation:

Everyday explanation:
In a "perfect" bell curve, the middle of the hill, the point with the most values, and the point that splits the students in half – are all in the same place.

Mathematical explanation:
In a symmetric normal distribution:
μ = median = mode
That is, the mean, median, and mode are all equal and located at the center of the distribution.

Question 4

2.50 pts

📊 What is standard deviation?
Intuitive explanation: what does the standard deviation "measure" in a distribution?

How much the data are spread around the mean – on average, how far they are from it.

The total number of data points in the sample.

The largest value minus the smallest value.

The size of the x-axis on the graph.

Explanation:

Everyday explanation:
The standard deviation tells us how "spread out" the scores are. If the standard deviation is small – most students are close to one another and to the mean. If it is large – there is wide spread, with very high and very low scores.

Mathematical explanation:
The standard deviation is the square root of the average squared deviations from the mean. It measures the spread around the mean, in units of the variable itself.

Question 5

2.50 pts

📈 Comparing spread:
Students in two classes on the same test have a mean score of 80. In class A the standard deviation is 3, and in class B the standard deviation is 12. What is true?

The scores in class B are spread out much more than the scores in class A.

The scores in class A are spread out more, because its standard deviation is smaller.

In both classes the spread is identical because the mean is identical.

You cannot compare spread using standard deviation.

Explanation:

Explanation:
A larger standard deviation = larger spread. In class B the standard deviation is 4 times larger (12 vs. 3), so the scores there are farther from the mean – there are more very strong and very weak students.

Question 6

2.50 pts

🧮 What is a z-score?
Choose the correct definition for a z-score.

A z-score is how many standard deviations a value is above or below the mean.

A z-score is the original score divided by 100 to convert it into a percentage.

A z-score is the mean of all the scores in the class.

A z-score is the difference between the highest and lowest scores.

Explanation:

Everyday explanation:
A z-score tells us "how far am I from the mean in units of standard deviations". For example, z = 2 means: I am two standard deviations above the mean.

Mathematical explanation:
A z-score is calculated as:
z = (X - μ) / σ
That is, the difference between the value and the mean, divided by the standard deviation.

Question 7

2.50 pts

➕➖ because :
z-score z ?

find belowmean.

find mean.

equal mean.

the value does not exist in the distribution

Explanation:

(X - μ) , X small mean, z . mean .

Question 8

2.50 pts

⚠️ Error identification in z-score:
Given a test score X=90, mean μ=80, standard deviation σ=5.
Or claimed: "My z-score is z = 5 because 90 - 80 = 10 and that is twice the standard deviation".
What is the problem with the claim?

She forgot to divide by the standard deviation; the true z-score is z = (90-80)/5 = 2.

She should have subtracted 5 from the mean rather than from the score.

She should have divided 80 by 90.

She was right, z=5 is correct.

Explanation:

Common mistake: Thinking that "twice the standard deviation" simply gives z=5 because the standard deviation is 5. In fact we do not divide by 5.

Correct calculation:
z = (X - μ) / σ = (90 - 80) / 5 = 10 / 5 = 2.

Question 9

2.50 pts

📑 – :
distribution normal ( Z), \(P(Z > 0)\) ?

0.5000

0.0000

1.0000

0.2500

Explanation:

normal (μ=0, σ=1) 0. Yes area find -0 .

:
P(Z > 0) = 0.5.

Question 10

2.50 pts

📑 : P(Z > 1.0)
, \(P(Z > 1.0)\) ?

0.1587

0.3413

0.8413

0.5000

Explanation:

No Z P(Z > z) (area ). z=1.0 0 0.1587.

probability large mean 15.87%.

Question 11

2.50 pts

📑 : P(Z > 1.5)
\(P(Z > 1.5)\) ?

0.0668

0.4332

0.9332

0.5000

Explanation:

: z = 1.5, area 0.0668, -6.7% area.

Question 12

2.50 pts

🔄 :
given \(P(Z > z)\) z .
calculate \(P(Z < -1.2)\) ?

: P(Z < -1.2) = P(Z > 1.2) 1.2 .

calculate 1 - P(Z > 1.2) because always -1.

1.2 -2.

calculate because because .

Explanation:

because Z , distribution . Yes z equal :

P(Z < -1.2) = P(Z > 1.2).

– because area equal.

Question 13

2.50 pts

⚠️ :
z=1.0 0.8413 0.1587 \(P(Z > 1.0)\).
?

P(Z < 1.0) P(Z > 1.0).

Z .

No (z=0.1).

No z=1.0.

Explanation:

0.8413 area (P(Z < 1.0)), 0.1587.

between "area " between "area " No .

Question 14

2.50 pts

🔁 :
given : \(P(Z > 1.2) = 0.1151\) ().
\(P(Z < 1.2)\) ?

0.8849

0.1151

0.0576

0.5000

Explanation:

area below 1. Yes:
P(Z < 1.2) = 1 - P(Z > 1.2) = 1 - 0.1151 = 0.8849.

Question 15

2.50 pts

🧮 z:
mean 70 8. 86.
?

2.0

1.0

1.5

-2.0

Explanation:

calculate:

z = (X - μ) / σ = (86 - 70) / 8 = 16 / 8 = 2.

find mean.

Question 16

2.50 pts

📉 mean:
mean 80, 10. 65.
?

-1.5

-1.0

1.5

0.15

Explanation:

z = (65 - 80) / 10 = -15 / 10 = -1.5.

find belowmean.

Question 17

2.50 pts

🎨 area -z:
.
probability ?

probability z₀: P(Z > z₀).

probability z₀: P(Z < z₀).

area below: P(Z ) = 1.

area between because Z.

Explanation:

find z₀, Yes -P(Z > z₀).

Question 18

2.50 pts

🎨 area between because:
between because z₁ -z₂.
probability ?

P(z₁ < Z < z₂)

P(Z > z₂)

P(Z < z₁)

P(Z > z₁) + P(Z > z₂)

Explanation:

find between because z₁ -z₂, Yes probability -Z between because : P(z₁ < Z < z₂).

Question 19

2.50 pts

📏 area -Z:
find because P(Z > z) ≈ 0.1587.
z ?

1.0

0.5

2.0

1.5

Explanation:

distribution normal 0.1587 -z=1.0 ( ).

Question 20

2.50 pts

📏 :
P(Z > z) ≈ 0.0228. z ?

2.0

1.0

3.0

0.2

Explanation:

: z≈2.0 P(Z > 2.0) ≈ 0.0228. as small more – z more mean.

Question 21

2.50 pts

📚 68%:
normal distribution, area find between z=-1 -z=1 ?

-68%

-50%

-95%

-99.7%

Explanation:

: 68% because find standard deviation mean (-z=-1 z=1).

Question 22

2.50 pts

📚 95%:
area find between z=-2 -z=2 ?

-95%

-68%

-50%

-99.7%

Explanation:

68–95–99.7 : -95% because find mean.

Question 23

2.50 pts

📚 99.7%:
area find between z=-3 -z=3 ?

-99.7%

-95%

-68%

-75%

Explanation:

-99.7% because find mean.

Question 24

2.50 pts

🧠 X -Z:
because 10. 90 z=1. because standard deviation, 100 z=2.
Correct?

because 100 more mean 90 because .

because 90 more mean 100 because .

both scores are at the same distance from the mean

equal z.

Explanation:

z . z=2 mean z=1, Yes 100 more mean 90.

Question 25

2.50 pts

👨‍👩‍👧 :
normal mean 170 cm standard deviation 6 cm. 182 cm – z ?

2.0

1.0

-2.0

0.2

Explanation:

z = (182 - 170) / 6 = 12 / 6 = 2.

mean.

Question 26

2.50 pts

👨‍👧 182 cm:
": μ=170, σ=6. find/ probability -182 cm.

P(Z > 2) ≈ 0.0228

P(Z > 1) ≈ 0.1587

P(Z < 2) ≈ 0.9772

P(Z < 0) = 0.5

Explanation:

calculate z=2. No Yes :
P(Z > 2) ≈ 0.0228.

Question 27

2.50 pts

📘 92:
normal: μ=80, σ=8. z 92, P(X > 92)?

z=1.5 probability 0.0668

z=1.0 probability 0.1587

z=2.0 probability 0.0228

z=0.5 probability 0.3085

Explanation:

z = (92 - 80) / 8 = 12/8 = 1.5.

: P(Z > 1.5) ≈ 0.0668.

Question 28

2.50 pts

⚠️ : 1
P(Z > 1.2). find 0.8849 . ?

P(Z < 1.2) P(Z > 1.2); 1 - 0.8849.

he should have looked up z = −1.2

he should have divided the value by 2

– 0.8849 .

Explanation:

0.8849 area -1.2. 1 - 0.8849 = 0.1151. : No Correct 1 .

Question 29

2.50 pts

📍 z=0?
normal distribution , z=0 -x?

mean distribution.

the standard deviation of the distribution

the maximum possible value

-Z (z=0.5).

Explanation:

mean 0 1. Yes z=0 mean – .

Question 30

2.50 pts

📐 area below:
area below distribution normal?

probability – 1.

– .

area – .

area .

Explanation:

probability, area below always equal -1 "100% ". area .

Question 31

2.50 pts

🏫 more normal?
more normal distribution?

Heights of students in the same year group.

Heights of adults in different age groups.

Shoe sizes of students in the same year group.

Number of hours of sleep per night.

Explanation:

log , Yes normal. , distribution No normal.

Question 32

2.50 pts

🚩 "":
z-score calculate "" more?

z=3

z=1

z=0.5

z=-1

Explanation:

as z large more, more mean calculate more. z=3 more .

Question 33

2.50 pts

📉 area small :
P(Z > z) small ( 0.001), Correct z?

z large – mean .

z small – mean.

z No.

z always equal -0.

Explanation:

area small because mean. z large ( 3 ).

Question 34

2.50 pts

⚠️ Z:
Z P(Z < z), P(Z > z). ?

between (large small).

– always P(Z < z) = P(Z > z).

always 0.5 z.

No calculate probability.

Explanation:

No , – area small area large . between .

Question 35

2.50 pts

📊 area between because:
P(Z > z). P(-1 < Z < 1), Correct?

calculate 1 - 2·P(Z > 1)

calculate 2·P(Z > 1)

calculate 1 - P(Z > 1)

calculate .

Explanation:

:
P(-1 < Z < 1) = 1 - 2·P(Z > 1).

Question 36

2.50 pts

🧾 :
given -X normal. probability between μ-σ between μ+σ.
?

probability 68%.

probability 50% because "".

probability 95%.

No .

Explanation:

68–95–99.7 : standard deviation mean 68% because.

Question 37

2.50 pts

⚠️ standard deviation:
: " 10, Yes find between 70 -80".
given mean 75. No Correct ?

standard deviation standard deviation ( Correct 65 85).

standard deviation mean 75.

Correct 0 100 always.

10 mean.

Explanation:

σ=10 -μ 75, standard deviation 65 85 (μ-σ μ+σ). .

Question 38

2.50 pts

⚠️ between X -Z:
-z=2. : " 2". ?

z=2 z-score, No ; -X between .

z between 0 -1 Yes 2 .

z=2 always 100.

– z -X .

Explanation:

z=2 mean . :
X = μ + z·σ.

Question 39

2.50 pts

🎨 :
normal distribution mean. more , "" more.
standard deviation large more?

more – large more.

more – .

No standard deviation.

cannot be determined from the graph

Explanation:

standard deviation large – "" more .

Question 40

2.50 pts

✅ Correct:
normal distribution ?

normal distribution mean 0 1.

normal distribution mean 1 0.

normal distribution distribution mean equal .

normal distribution z large -0.

Explanation:

, normal distribution N(0,1): mean 0 standard deviation 1. normal distribution "" z-score.