Averages and data displays

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What is an Average?

An average is a single number that stands for a whole group of numbers.
It tells you the center or the most typical value in a set of data.
There are three common averages: mean, median, and mode.
The range is not an average, but it tells you how spread out the numbers are.
Different averages are useful in different situations.

The Mean

The mean is the most common average. It is sometimes called the arithmetic mean.
To find the mean, add up all the numbers, then divide by how many numbers there are.
Example: For the numbers 2, 3, 4, 7, 9, the mean is (2+3+4+7+9) ÷ 5 = 25 ÷ 5 = 5.
The mean is like sharing the total equally among all the items.
If one number is very large or very small, it can change the mean a lot.

The Median

The median is the middle number when you put the numbers in order from smallest to largest.
To find the median, first sort the numbers. If there is an odd number of values, the median is the middle one.
If there is an even number of values, the median is the mean of the two middle numbers.
Example: For 2, 3, 4, 7, 9, the median is 4.
The median is good to use when there are outliers (very high or very low numbers) because it is not pulled by them.

The Mode

The mode is the number that appears most often in a set of data.
A set of numbers can have one mode, more than one mode, or no mode at all.
Example: In 2, 3, 3, 5, 3, the mode is 3 because it appears three times.
The mode is useful for categorical data (like favorite colors) where numbers are labels, not amounts.
The mode tells you the most popular or common choice.

The Range

The range is the difference between the largest and the smallest number in a set.
To find the range, subtract the smallest number from the largest number.
Example: For 2, 3, 4, 7, 9, the range is 9 − 2 = 7.
The range tells you how spread out the data is.
A small range means the numbers are close together; a large range means they are far apart.

Choosing the Best Average

The mean is best when the numbers are fairly even and there are no extreme values.
The median is best when there are outliers (very high or very low numbers) because it is not affected by them.
The mode is best when you want to know the most common value, especially for categories.
Example: For house prices, the median is often used because a few very expensive houses would make the mean too high.
Always think about what you want to know before picking an average.

Reading and Drawing Data Displays

A dot plot shows each data point as a dot above a number line. It is good for small sets of data.
A bar chart uses bars to show how many times each category or value occurs.
A pie chart shows parts of a whole as slices of a circle.
A line graph shows how data changes over time, with points connected by lines.
A histogram is like a bar chart but for continuous data, where bars touch each other.

Describing Data from a Graph

When you look at a graph, describe the shape: is it symmetric or skewed?
Spread means how much the data varies. You can see it by looking at the range or how wide the graph is.
Look for clusters (groups of data close together) and gaps (empty spaces).
Look for outliers – points that are far away from the rest.
Use the graph to answer questions like: What is the most common value? What is the range?

Misleading Graphs

A graph can be misleading if it is not drawn fairly.
Watch out for broken axes (where the scale does not start at zero) – it can make differences look bigger.
Watch out for uneven scales (where the intervals are not equal) – it can distort the data.
Watch out for 3D effects or pictures that make one bar look bigger than it really is.
Always check the labels and the scale before you trust a graph.

Finding a Missing Value

If you know the mean and the number of values, you can find the total sum.
To find the total sum, multiply the mean by the number of values: total = mean × count.
If you know the total sum and all but one value, you can find the missing value by subtracting the known values from the total.
Example: The mean of 5 numbers is 6, so the total is 5 × 6 = 30. If four numbers are 4, 5, 7, 8 (sum = 24), the missing number is 30 − 24 = 6.
This method works for any set of numbers.

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1.A statistical question is a question that can be answered by collecting data and where the answer is expected to vary.
Easy
True or false?
2.What is the mean (average) of the numbers 4, 8, 6, and 10?
Easy
- A7
- B6
- C8
- D5
3.What is the mode of the data set: 5, 7, 5, 3, 5, 8?
Easy
- A5
- B7
- C3
- D8
4.The mean of five numbers is 6. Four of the numbers are 4, 8, 5, and 7. What is the fifth number?
Medium
- A6
- B5
- C8
- D9
5.The median is always the same as the mean.
Easy
True or false?
6.Which measure of center is best to represent the typical income of a town where a few people are extremely rich?
Medium
- AMean
- BMedian
- CMode
- DRange
7.Which of the following are measures of central tendency? (Select all that apply)
Medium
- AMean
- BMedian
- CMode
- DRange
- ESum
8.Arrange the steps to find the median of a data set in the correct order.
Medium
- Sort the numbers from smallest to largest.
- If there is an odd number of values, pick the middle one.
- If there is an even number of values, average the two middle ones.

Thêm câu hỏi

9.Match each term with its correct definition.
Medium
- Mean
- Median
- Mode
- Range
- The sum divided by the count
- The middle value after sorting
- The value that occurs most often
- The difference between the largest and smallest
10.What is the median number of pets?
Medium
- A1
- B2
- C0
- D3
11.The sum of the deviations from the mean is always zero.
Easy
True or false?
12.Which of the following is NOT a measure of central tendency?
Easy
- AMean
- BMedian
- CMode
- DRange
13.A student's scores on four tests are 70, 80, 90, and 100. What score would they need on a fifth test to have a mean of 85?
Medium
- A85
- B80
- C90
- D75
14.The mode can be used for categorical data, such as favorite colors.
Easy
True or false?
15.Which of the following are true about the mean? (Select all that apply)
Medium
- AIt is the sum of the values divided by the count.
- BIt is always one of the data values.
- CIt is affected by outliers.
- DIt is also called the arithmetic average.
16.Put the following steps in order to calculate the mean of a data set.
Medium
- Add all the numbers together.
- Divide the sum by the number of numbers.
- Write down the sum.
17.Match each graph type with its typical use.
Medium
- Bar chart
- Line graph
- Pie chart
- Compare categories
- Show change over time
- Show parts of a whole
18.If you multiply every number in a data set by 2, the mean also doubles.
Medium
True or false?
19.Match each measure of central tendency with its correct definition.
Easy
- Mean
- Median
- Mode
- The sum of all values divided by the number of values
- The middle value when the data is arranged in order
- The value that appears most often
20.Put these steps in order to find the median of a data set.
Easy
- Sort the numbers from smallest to largest.
- If there is an odd number of values, pick the middle one.
- If there is an even number of values, find the mean of the two middle numbers.
21.Which of the following are true about the median? (Select all that apply)
Medium
- AIt is the most common value.
- BIt is the middle value when data is sorted.
- CIt is affected by outliers.
- DIt is not affected by extreme values.
- EIt can be found for categorical data.
22.The mode can be used with categorical data, such as favorite colors.
Easy
True or false?
23.A class has test scores: 60, 70, 70, 80, 90. What is the median score?
Medium
- A70
- B75
- C80
- D70.5
24.The mean of a data set is 12. If each value in the data set is multiplied by 3, what is the new mean?
Medium
- A12
- B36
- C15
- D4
25.If a data set has an outlier, the median is usually a better measure of central tendency than the mean.
Medium
True or false?
26.Place the steps in order to find the mean of a data set.
Medium
- Add all the numbers together.
- Count how many numbers there are.
- Divide the sum by the count.
27.What is the mode of the numbers: 3, 5, 5, 7, 9?
Easy
- A3
- B5
- C7
- D9
28.The mean of four numbers is 15. Three of the numbers are 10, 20, and 12. What is the fourth number?
Hard
- A18
- B15
- C17
- D20
29.The mean is always one of the numbers in the data set.
Easy
True or false?
30.Match each scenario with the best measure of central tendency.
Medium
- Typical income in a town with a few very rich people
- Most common shoe size
- Average score on a test with no outliers
- Median
- Mode
- Mean
31.Put these steps in order to create a dot plot from a data set.
Medium
- Draw a number line that includes all values.
- Place a dot above the number line for each data value.
- Label the number line and give the plot a title.
32.What is the range of the numbers: 2, 8, 3, 10?
Easy
- A8
- B7
- C10
- D5
33.A data set has numbers 2, 4, 6, 8, 10. If each number is increased by 3, what is the new mean?
Medium
- A8
- B9
- C6
- D7
34.The range is a measure of central tendency.
Medium
True or false?
35.Match each term with its definition.
Medium
- Outlier
- Skewed data
- Symmetric data
- A value far away from the other values
- Data that has a long tail on one side
- Data that is evenly spread around the center
36.Put these steps in order to find the median of the data set: 4, 1, 7, 3, 9.
Medium
- Sort the numbers: 1, 3, 4, 7, 9.
- Count the numbers: 5.
- Pick the middle number: 4.
37.Which of the following are true about the mode? (Select all that apply)
Medium
- AIt can be used with categorical data.
- BIt is always a number.
- CThere can be more than one mode.
- DIt is the middle value.
38.What is the mode of the numbers: 4, 4, 5, 6, 6, 6, 7?
Medium
- A4
- B5
- C6
- D7
39.What is the median of the numbers 3, 1, 4, 1, 5?
Easy
- A3
- B1
- C4
- D5
40.What is the mode of the numbers 2, 2, 3, 4, 4, 4, 5?
Easy
- A4
- B2
- C3
- D5
41.Find the median of this set of numbers: 12, 3, 7, 9, 15. Type your answer as a number.
Medium
42.Calculate the range of the numbers: 2, 9, 4, 7, 12. Type your answer.
Medium
43.Fill in the blank with the correct word.
Easy
The ____ is the value that appears most often in a set of data.
44.What is the mean of the numbers 2, 4, 6, and 8? Write your answer as a number.
Easy
45.How many students are in the class?
Medium
46.What is the mode of the number of pets?
Medium
47.How many students read at least 2 books?
Medium
48.What is the mean number of books read per student? Round to the nearest tenth if needed.
Medium
49.Fill in the blank: The arithmetic mean is calculated by dividing the ___ of the numbers by the count.
Easy
The arithmetic mean is calculated by dividing the ____ of the numbers by the count.
50.Find the median of the numbers: 5, 2, 8, 3, 7, 10. Type your answer.
Medium
51.What is the mode of the set: 3, 7, 3, 9, 3, 7?
Easy
52.Fill in the blank: The ___ is the value in the middle after sorting the data.
Easy
The ____ is the value in the middle after sorting the data.
53.Explain why the median is often a better measure of central tendency than the mean when a data set has an outlier.
Hard
54.Find the median of this set of numbers: 12, 3, 7, 9, 15. Type your answer as a number.
Medium
- A9
- B11
- C8
- D10
55.Calculate the range of the numbers: 2, 9, 4, 7, 12. Type your answer.
Medium
- A11
- B9
- C10
- D12
56.How many students are in the class?
Medium
- A13
- B10
- C11
- D12
57.What is the mode of the number of pets?
Medium
- A0 pets
- B2 pets
- C1 pets
- D3 pets
58.How many students read at least 2 books?
Medium
- A6 students
- B9 students
- C8 students
- D7 students
59.What is the mean number of books read per student? Round to the nearest tenth if needed.
Medium
- A3.4 books
- B0.4 books
- C2.4 books
- D1.4 books
60.Find the median of the numbers: 5, 2, 8, 3, 7, 10. Type your answer.
Medium
- A7
- B5
- C6
- D8

Ghi chú ôn tập

## Averages and Data Displays – Quick Revision ### What is an Average? An **average** is a single value that stands for a whole group of numbers. It tells you the "center" or "typical" value. The three most common averages are the **mean**, **median**, and **mode**. You also need the **range** to describe how spread out the data is. ### The Mean - Add up all the numbers, then divide by how many numbers there are. - Formula: mean = (sum of all values) / (number of values) - Example: For 2, 3, 4, 7, 9 → sum = 25, count = 5 → mean = 25 ÷ 5 = 5 - Use the mean when the data is fairly even and has no very high or very low values. ### The Median - Put the numbers in order from smallest to largest. - The median is the middle number. - If there are two middle numbers, the median is the mean of those two. - Example: For 2, 3, 4, 7, 9 → median = 4 - Use the median when there are extreme values (outliers) because it is not pulled by them. ### The Mode - The mode is the value that appears most often. - A set can have one mode, more than one mode, or no mode. - Example: For 2, 3, 3, 4, 7 → mode = 3 - Use the mode for categorical data (like favourite colour) or to find the most common value. ### The Range - The range is the difference between the largest and smallest values. - Formula: range = largest value − smallest value - Example: For 2, 3, 4, 7, 9 → range = 9 − 2 = 7 - A large range means the data is spread out; a small range means it is close together. ### Data Displays - **Dot plot**: A number line with dots above each value. Shows how often each value occurs. - **Bar chart**: Uses bars to show counts for categories. Bars do not touch. - **Pie chart**: A circle divided into slices to show parts of a whole. - **Line graph**: Shows change over time by connecting points with lines. - **Histogram**: Like a bar chart but for numbers grouped into intervals. Bars touch. ### Shape and Spread - Look at the overall shape: where is the peak? Is it symmetric or skewed? - Spread tells you how varied the data is. Use the range and the shape to describe it. ### Misleading Displays - Be careful! Graphs can trick you if the scale starts at a number other than zero, or if the bars are not the same width. - Always check the labels and the scale. ### Finding a Missing Value - If you know the mean and the number of values, you can find the total sum: sum = mean × count. - Then subtract the known values to find the missing one. - Example: Mean of 5 numbers is 6, so sum = 5 × 6 = 30. If four numbers are 4, 5, 7, 8 (sum = 24), the missing number is 30 − 24 = 6.

Average: A single value that represents a group of numbers. The mean, median, and mode are all averages.
Mean: The sum of all values divided by the number of values. It is the most common average.
Median: The middle value when numbers are arranged in order. If there are two middle numbers, it is the mean of those two.
Mode: The value that appears most often in a set of data.
Range: The difference between the largest and smallest values in a data set.
Outlier: A value that is much higher or much lower than the other values in a data set.
Dot plot: A display that uses dots above a number line to show how often each value occurs.
Bar chart: A display that uses bars to show counts for different categories. Bars do not touch.
Pie chart: A circular display that shows parts of a whole as slices.
Line graph: A display that shows change over time by connecting points with lines.
Histogram: A display that uses bars to show the frequency of numbers grouped into intervals. Bars touch.
Statistical question: A question that expects an answer that varies, based on data collected from a group.

Thẻ ghi nhớ

What is a statistical question?

A statistical question is a question that can be answered by collecting data and where the answer will vary. For example, 'How tall are the students in our class?' is statistical because heights vary.

What is the mean?

The mean is the average found by adding all the numbers and dividing by how many numbers there are. For example, the mean of 2, 3, and 7 is (2+3+7)/3 = 12/3 = 4.

How do you find the median?

The median is the middle number when the numbers are put in order from smallest to largest. If there are two middle numbers, the median is the mean of those two.

What is the mode?

The mode is the number that appears most often in a set of data. A set can have one mode, more than one mode, or no mode if all numbers appear the same number of times.

What is the range?

The range is the difference between the largest and smallest numbers in a set. To find it, subtract the smallest from the largest.

What is the mean of the numbers 5, 8, 9, 12?

The mean is 8.5 because (5+8+9+12)/4 = 34/4 = 8.5.

What is the median of the numbers 3, 7, 2, 9, 5?

First put them in order: 2, 3, 5, 7, 9. The middle number is 5, so the median is 5.

What is the mode of the numbers 4, 6, 4, 7, 4, 8?

The mode is 4 because it appears three times, more than any other number.

What is the range of the numbers 12, 7, 20, 5?

The range is 15 because the largest is 20 and the smallest is 5, and 20 - 5 = 15.

When is the median a better average than the mean?

The median is better when there are outliers (very high or low numbers) because it is not affected by them. For example, median income is used because a few very rich people would make the mean too high.

What is a dot plot?

A dot plot is a graph that shows each data value as a dot above a number line. It helps you see how often each value occurs.

What is a bar chart?

A bar chart uses bars of different heights to show the number of items in each category. The bars do not touch each other.

What is a pie chart?

A pie chart is a circle divided into slices that show how a whole is split into parts. Each slice is a percentage of the total.

What is a line graph?

A line graph uses points connected by lines to show how something changes over time. It is good for trends.

What is a histogram?

A histogram is like a bar chart but for numbers grouped into intervals. The bars touch each other because the intervals are continuous.

How do you find a missing value when you know the mean?

Multiply the mean by the number of values to get the total sum. Then subtract the sum of the known values to find the missing one.

The mean of 4 numbers is 6. Three of the numbers are 4, 7, and 8. What is the fourth number?

The total sum is 4 × 6 = 24. The known sum is 4+7+8 = 19. So the missing number is 24 - 19 = 5.

What does the shape of a data display tell you?

The shape shows if the data is spread out evenly, clustered in the middle, or skewed to one side. For example, a symmetric shape means the data is balanced.

What does 'spread' mean in data?

Spread is how far apart the data values are. The range is one way to measure spread.

What is an outlier?

An outlier is a value that is much higher or much lower than the rest of the data. It can affect the mean but not the median.

How can a graph be misleading?

A graph can be misleading if it starts the scale at a number other than zero, or uses uneven intervals, making differences look bigger or smaller than they really are.

What is the mean of 10, 20, 30, 40, 50?

The mean is 30 because (10+20+30+40+50)/5 = 150/5 = 30.

What is the median of 1, 2, 3, 4, 5, 6?

The median is 3.5 because the two middle numbers are 3 and 4, and (3+4)/2 = 3.5.

What is the mode of the numbers 2, 2, 3, 4, 4, 4, 5?

The mode is 4 because it appears three times, more than any other number.

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