Worksheet 1 on Standard Deviation and Z-scores
1) 500 juniors at Central High School took the ACT last year. The scores were distributed normally with a mean of 24 and a
standard deviation of 4. Draw and label a bell curve.
2)
3)
4)
5)
a) What percentage of scores are between 20 and 28?
b) What percentage of scores are below 12?
c) What is the percentile (no calculator) of someone who scored a 32?
d) Approximately how many juniors scored below a 20?
e) Approximately how many juniors scored between a 24-32?
f) Between what scores are within one standard deviation from the mean?
g) Between what scores did 95% of the students score?
h) What is the percentile of someone who scored a 26? (use calculator)
i) What percentage of students scored between a 23 and 27? (use calculator)
j) How many students scored between a 21 and 24? (use calculator)
k) What score has a Z-score of -1?
The average on a test is 88 and the standard deviation is 3. The teacher then gives 4 points to every student. What will
the average become?
What will the standard deviation become?
Which of the following has the highest standard deviation?
A. Ages of people in a senior citizen home
B. Ages of people at an amusement park.
C. Ages of college freshmen
D. Ages of people at the supermarket.
The following are test scores: 100, 95, 93, 91, 83, 87, 84, 82, 84, 73, 72, and 71. Calculate the standard deviation and the
mean.
Assuming the data is normally distributed, calculate the percentage of students who got between a 80 and 90
(use calculator)
Suppose that your math teacher determined to give out grades based on the bell curve. Do you agree or disagree with that
decision? Why?
6)
A popular band on tour played a series of concerts in large venues. They always drew a large crowd averaging 21,365 fans.
Which of the following values is more likely to be correct for standard deviation? 20, 200, 2000 or 20,000?
7) The mean score on a test was 75 with a standard deviation of 5 points. Jack’s Z score was -2. How many points did he
score?
8) A high school student took her mid-term exams for French and Mathematics. In French, she scored 82 and in math an 86.
The overall results on the French exam had a mean of 72 and a standard deviation of 8 while the mean math score was 68
with a standard deviation of 12. On which exam did she do better compared to the other students who took the same
exam? (explain)
9) A town’s January high temperature averages 36oF with a standard deviation of 10o while in July the mean high temperature
is 74o and the standard deviation is 8o. In which month is it more unusual to have a day with a high temperature of 55 o?
(explain)
10) A score of 73 is two standard deviations below the mean. A score of 88 is one standard deviation above the mean. Find the
mean and the standard deviation.
11) A temperature of 88 degrees is one standard deviation below the mean. A temperature of 98 is three standard deviations
above the mean. Find the mean and standard deviation.
12) Give two examples of ages of 10 people so that the mean is 25 for both groups but the standard deviation is very low for
one group and very high for the other group. Find the standard deviation of both groups.
13) Are the following situations skewed negatively (N), positively (P) or normally distributed?
A. The weights of new born babies
B. Household income
C. The number of siblings that a person has
D. The ages of people in your math class