Histograms

Show your work for each problem.

1.
What makes a histogram different from a bar graph?
1. a)Bars are different colors
2. b)Bars are horizontal
3. c)Bars touch each other
4. d)Bars are taller
Answer: Bars touch each other
In a histogram, bars touch each other because the data is continuous. There are no gaps between intervals since the end of one interval is the beginning of the next.
2.
A histogram shows test scores. The interval has a bar height of . What does this mean?
1. a)Students scored 8 points each
2. b)8 is the highest score
3. c)The average score is 8
4. d)8 students scored between 60 and 69
Answer: 8 students scored between 60 and 69
The bar height of represents the frequency - the number of students who scored in that interval. So 8 students scored between 60 and 69 (the interval excludes 70).
3.
A histogram has 3 bars with frequencies , , and . What is the total number of data points?
Answer: 24
The total number of data points equals the sum of all frequencies: data points.
4.
Which type of data is best displayed using a histogram?
1. a)Names of countries
2. b)Types of pets
3. c)Favorite pizza toppings
4. d)Heights of students in cm
Answer: Heights of students in cm
Heights of students (in cm) are continuous numerical data that can be grouped into intervals like 150-160 cm, 160-170 cm, etc. The other options are categorical data better suited for bar graphs.
5.
In a histogram, the intervals are: , , . What is the width of each interval?
Answer: 10
The interval width is the difference between the upper and lower bounds: , , . Each interval has a width of 10.
6.
A histogram shows ages of visitors: 0-10 (15), 10-20 (28), 20-30 (32), 30-40 (18), 40-50 (7). How many visitors are under 30 years old?
Answer: 75
Visitors under 30 are in intervals 0-10, 10-20, and 20-30. Adding: visitors are under 30 years old.
7.
A value of needs to be placed in a histogram. The intervals are , , . Which interval should it go in?
1. a)10-20
2. b)30-40
3. c)20-30
4. d)None of these
Answer: 20-30
Since and , it falls in the interval . Remember: intervals include the lower bound but exclude the upper bound.
8.
Data ranges from 12 to 47. Create 5 equal intervals for a histogram.
Answer: 10-18, 18-26, 26-34, 34-42, 42-50
1. What is the range of the data? (maximum - minimum) 35
2. Divide the range by 5 to get the interval width 7
3. Round up to a nice number if needed. What width will you use? 8
4. Start at 10 (below 12). What is the first interval? 10-18
5. List all 5 intervals 10-18, 18-26, 26-34, 34-42, 42-50
9.
A histogram shows weights in kg: 50-60 (8), 60-70 (15), 70-80 (22), 80-90 (10), 90-100 (5). What percentage of data is in the 70-80 interval? Round to the nearest whole number.
Answer: 37
Total = . Percentage in 70-80 = , which rounds to .
10.
A histogram has most of its bars on the right side with a tail extending to the left. What is this distribution called?
1. a)Symmetric
2. b)Left-skewed (negatively skewed)
3. c)Right-skewed (positively skewed)
4. d)Uniform
Answer: Left-skewed (negatively skewed)
When the tail extends to the left, it's called left-skewed or negatively skewed. The name refers to the tail direction, not where most data is located.
11.
A histogram shows exam scores: 0-20 (3), 20-40 (7), 40-60 (18), 60-80 (15), 80-100 (7). Find the modal class (interval with highest frequency).
Answer: 40-60
1. List all the frequencies 3, 7, 18, 15, 7
2. Which frequency is the highest? 18
3. Which interval has this frequency? 40-60
12.
A histogram shows daily temperatures in °C: 10-15 (4), 15-20 (9), 20-25 (12), 25-30 (8), 30-35 (2). What is the estimated mean temperature? Use midpoints.
Answer: 21.8
1. Find the midpoint of each interval (e.g., midpoint of 10-15 is 12.5) 12.5, 17.5, 22.5, 27.5, 32.5
2. Multiply each midpoint by its frequency: 12.5×4, 17.5×9, 22.5×12, 27.5×8, 32.5×2 50, 157.5, 270, 220, 65
3. Sum all the products 762.5
4. Find the total frequency 35
5. Divide sum of products by total frequency 21.8
13.
A histogram shows running times in minutes: 20-25 (6), 25-30 (14), 30-35 (18), 35-40 (9), 40-45 (3). How many runners finished in less than 35 minutes?
Answer: 38
Less than 35 minutes includes intervals 20-25, 25-30, and 30-35. Total = runners.
14.
Two histograms show test scores. Class A: 0-20 (5), 20-40 (15), 40-60 (20), 60-80 (8), 80-100 (2). Class B: 0-20 (2), 20-40 (6), 40-60 (12), 60-80 (18), 80-100 (12). Which class performed better and why?
Answer: Class B
1. What is the modal class for Class A? 40-60
2. What is the modal class for Class B? 60-80
3. How many students in Class A scored 60 or above? 10
4. How many students in Class B scored 60 or above? 30
5. Which class performed better? Class B
15.
A histogram shows the ages of people at a birthday party: most bars are on the left with one tall bar around 10-15 and a small bar far on the right around 70-80. What type of event is this likely describing?
1. a)A college graduation
2. b)A child's birthday party with a grandparent
3. c)A retirement party
4. d)A wedding
Answer: A child's birthday party with a grandparent
The histogram shows many people aged 10-15 (children) and one person aged 70-80 (elderly). This pattern fits a child's birthday party where friends are invited along with a grandparent.
16.
A histogram has 6 equal intervals. The frequencies are: , , , , , . What fraction of the total data falls in the middle two intervals? Express as a simplified fraction.
Answer: 15/28
Middle two intervals: . Total: . Fraction = .