Averages and range

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Notas de aula

What is an average?

An average is a single value that summarises the centre or typical value of a data set.
The most common average is the mean, but median and mode are also averages.
The three averages are often called the three Ms: mean, median and mode.
Averages are measures of central tendency — they show where data clusters.
The range is not an average; it measures how spread out the data is.

The mean

The mean is found by adding all values and dividing by how many there are.
For a list of n values, mean = sum of values ÷ n.
Example: for 2, 3, 4, 7, 9 the mean is (2+3+4+7+9) ÷ 5 = 25 ÷ 5 = 5.
The mean uses every value, so it can be pulled up or down by extreme values (outliers).

The median

The median is the middle value when the data is written in order.
If there is an odd number of values, the median is the single middle value.
If there is an even number of values, the median is the mean of the two middle values.
The median is useful when data is skewed or has outliers, because it is not affected by extreme values.
Example: personal income is often reported as the median so that a few very high incomes do not distort the typical value.

The mode

The mode is the value that appears most often in a data set.
A data set can have one mode, more than one mode, or no mode.
The mode is the only average that can be used for categorical (non-numeric) data.
The mode is useful when you want the most common or most frequent value.

The range

The range is the difference between the largest and smallest values: range = largest − smallest.
The range measures the spread of the data, not its centre.
A small range means the data values are close together; a large range means they are spread out.
The range is affected by outliers because it depends only on the extreme values.

Choosing the most appropriate average

Use the mean when the data is fairly symmetrical and you want to use all the values.
Use the median when the data is skewed or has outliers, so the typical value is not distorted.
Use the mode when the data is categorical or when the most common value is most useful.
The best average depends on the context and the purpose of the summary.

Finding a missing value when the mean is known

If you know the mean and the number of values, you can find the total: total = mean × number of values.
To find a missing value, subtract the sum of the known values from the total.
Example: if the mean of 4 numbers is 10, the total is 40. If three numbers are 8, 9 and 12 (sum 29), the missing number is 40 − 29 = 11.

Averages from a frequency table

In a frequency table, the mean is calculated as (sum of value × frequency) ÷ (sum of frequencies).
The median is the middle value when the data is listed in order, using the frequencies to count positions.
The mode is the value with the highest frequency.
The range is the largest value minus the smallest value in the table.

Averages from grouped data

For grouped data, you do not know the exact values, so you use the midpoint of each class interval.
The estimated mean is calculated as (sum of midpoint × frequency) ÷ (sum of frequencies).
The modal class is the class interval with the highest frequency.
The estimated mean is an approximation because it assumes all values in a class are at the midpoint.

Comparing two data sets

Compare two data sets by comparing an average (mean or median) and the range.
The average tells you which data set is typically higher or lower.
The range tells you which data set is more spread out or consistent.
A smaller range usually means the data is more consistent.

Slides

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Questões de prática

Prévia grátis — 8 de 62 perguntas. Cadastre-se para ver todas.

1.Which of these is the most common meaning of the word 'average' in mathematics?
Easy
- AThe arithmetic mean
- BThe harmonic mean
- CThe geometric mean
- DThe mid-range
2.What is the median of the data set: 2, 3, 4, 7, 9?
Easy
- A4
- B5
- C7
- D9
3.What is the mode of the data set: 2, 3, 4, 4, 7, 9?
Medium
- A2
- B4
- C5
- D9
4.What is the range of the data set: 2, 3, 4, 7, 9?
Medium
- A2
- B7
- C9
- D5
5.Which of the following are measures of central tendency? (Select all that apply)
Medium
- AMean
- BMedian
- CMode
- DRange
- EStandard deviation
6.The median is always one of the values in the data set.
Easy
True or false?
7.Match each term with its definition.
Medium
- Mean
- Median
- Mode
- Range
- The sum of values divided by the number of values
- The middle value when data is ordered
- The most frequent value
- The difference between the highest and lowest values
8.Match each term to its meaning.
Easy
- Mean
- Median
- Mode
- Range
- The sum of the values divided by how many there are
- The middle value when the data is put in order
- The value that occurs most often
- The difference between the largest and smallest values

Mais perguntas

9.Put these steps in the correct order for finding the median of a list of numbers.
Medium
- Write the numbers in order of size
- Count how many numbers there are
- If the count is odd, take the middle number
- If the count is even, find the mean of the two middle numbers
10.Which of the following statements are true about the mode? (Select all that apply)
Medium
- AIt can be used for categorical data such as eye colour
- BIt is the value that appears most frequently
- CIt is always one of the numbers in the data set
- DIt is affected by multiplying every value by 2
- EA data set can have more than one mode
11.A data set contains one very large outlier. Which average is usually most appropriate to represent the typical value?
Medium
- AThe median
- BThe mean
- CThe range
- DThe mid-range
12.The range is a measure of how spread out the data is.
Easy
True or false?
13.Which averages are always somewhere between the smallest and largest values in the data set? (Select all that apply)
Medium
- AMean
- BMedian
- CMode
- DRange
- EMid-range
14.Which of the following is a measure of central tendency?
Medium
- AMean
- BRange
- CStandard deviation
- DInterquartile range
15.A data set has a mean of 20 and a range of 5. Every value is multiplied by 3. What are the new mean and range?
Medium
- AMean 60, range 15
- BMean 60, range 5
- CMean 20, range 15
- DMean 23, range 8
16.Data set A has a mean of 30 and a range of 4. Data set B has a mean of 30 and a range of 12. Which statement best compares the two data sets?
Medium
- AThe two data sets have the same typical value but B is more spread out
- BThe two data sets have the same spread but B has a larger typical value
- CData set A has a larger typical value and is more spread out
- DThe two data sets are identical
17.Put these averages in order from smallest to largest for the data set 3, 4, 4, 6, 100.
Medium
- Mode
- Median
- Mean
18.What is the mean of the numbers 4, 6, 8, 10?
Easy
- A7
- B6
- C8
- D28
19.What is the range of the data set: 12, 7, 15, 9, 11?
Medium
- A8
- B15
- C7
- D3
20.The mean is always one of the values in the data set.
Easy
True or false?
21.Which of the following statements about the range are true? (Select all that apply)
Medium
- AThe range is a measure of spread.
- BThe range uses only the largest and smallest values.
- CThe range is always greater than the mean.
- DThe range is a measure of central tendency.
22.Put these steps in the correct order to find the median of a list of numbers.
Medium
- Write the numbers in order of size.
- If there is an even number of values, find the mean of the two middle values.
- If there is an odd number of values, choose the middle value.
23.What is the mode of the data set: 3, 5, 5, 5, 7, 9?
Medium
- A5
- B7
- C3
- D9
24.A data set has a mean of 24 and a range of 7. Every value in the data set is multiplied by 4. What are the new mean and the new range?
Medium
- Amean 96, range 28
- Bmean 96, range 7
- Cmean 28, range 28
- Dmean 24, range 28
25.The mean of six numbers is 9. A seventh number, 23, is added to the data set. What is the mean of all seven numbers?
Medium
- A11
- B9
- C10
- D12
26.A data set has a mean of 30 and a range of 10. Every value is divided by 2. Which statements are true? (Select all that apply)
Hard
- AThe new mean is 15.
- BThe new range is 5.
- CThe new mean is 60.
- DThe new range is 10.
- EThe new mean is 28.
27.The mean of the numbers 4, 7, x and 12 is 9. What is the value of x?
Medium
- A13
- B9
- C11
- D15
28.A data set has a mean of 25 and a range of 8. Every value is multiplied by 3 and then 5 is added to each result. What is the new range?
Hard
- A24
- B29
- C8
- D13
29.A data set has a mean of 40 and a range of 12. Every value is divided by 4. What is the new range?
Medium
- A3
- B48
- C8
- D12
30.A data set has a mean of 20 and a range of 10. Every value is multiplied by 2 and then 6 is subtracted from each result. What statement is true?
Hard
- AThe new mean is 34 and the new range is 20.
- BThe new mean is 40 and the new range is 10.
- CThe new mean is 34 and the new range is 10.
- DThe new mean is 40 and the new range is 20.
31.The mean of five numbers is 14. The smallest number is removed, and the mean of the remaining four numbers is 16. What is the smallest number?
Hard
- A6
- B10
- C4
- D8
32.A data set consists of seven numbers. The sum of all seven numbers is 84. The smallest number is 6 and the largest number is 22. What is the range of this data set?
Hard
- A16
- B22
- C28
- D6
33.What is the most common meaning of the word 'average' in mathematics?
Easy
- Athe arithmetic mean
- Bthe range
- Cthe difference between the largest and smallest values
- Dthe number of values in the data set
34.Which formula gives the arithmetic mean of a list of n values x₁, x₂, ..., x_n?
Medium
- A\bar{x} = \frac{x₁ + x₂ + \cdots + x_n}{n}
- B\bar{x} = \frac{n}{x₁ + x₂ + \cdots + x_n}
- C\bar{x} = x_n - x₁
- D\bar{x} = \frac{x₁ + x_n}{2}
35.Which symbol is typically used to denote the arithmetic mean of a sample?
Easy
- A\bar{x}
- B\mu
- Cn
- Dr
36.The range is a measure of central tendency.
Easy
True or false?
37.What is the mean of the data set: 2, 3, 4, 7, 9?
Medium
- A5
- B4
- C25
- D6
38.Which of the following statements about the mean are true? (Select all that apply)
Medium
- AIt is calculated by adding all values and dividing by how many there are.
- BIt is always one of the values in the data set.
- CIt can be affected by outliers.
- DIt can only be used for numeric data.
- EIt is a measure of spread.
39.A data set has a mean of 20 and a range of 6. Every value is multiplied by 3. What are the new mean and the new range?
Medium
- Amean 60, range 18
- Bmean 60, range 6
- Cmean 23, range 9
- Dmean 20, range 18
40.A data set has a mean of 30 and a range of 10. Every value is divided by 2. What are the new mean and the new range?
Medium
- Amean 15, range 5
- Bmean 15, range 10
- Cmean 30, range 5
- Dmean 28, range 8
41.What is the term for the value that appears most frequently in a data set?
Easy
42.Complete the sentence about the range.
Easy
The range is the difference between the largest and the ____ values in a data set.
43.Calculate the mean of the following data set: 2, 3, 4, 7, 9.
Medium
44.The mean of five numbers is 10. Four of the numbers are 8, 9, 11, and 12. What is the fifth number?
Medium
45.In the grouped frequency table, which class is the modal class?
Easy
46.Complete the sentence with the correct word.
Medium
When a data set contains outliers, the ____ is often a better average to use than the mean.
47.Find the mean of the data set: 3, 5, 7, 9, 11.
Medium
48.The mean of five numbers is 12. Four of the numbers are 10, 14, 9, and 15. Find the fifth number.
Medium
49.Complete the sentence about the mode.
Easy
The mode is the value that appears ____ often in a data set.
50.A data set has a mean of 15 and a range of 6. Every value is multiplied by 2. What is the new mean?
Medium
51.The mean of five numbers is 12. Four of the numbers are 9, 11, 14 and 16. Find the fifth number.
Medium
52.A data set contains the values 4, 7, 7, 9, 13, 18. Find the range of the data set.
Medium
53.Explain why the median is often used instead of the mean when a data set contains outliers.
Hard
54.Explain why the harmonic mean is used to average speeds measured over the same distance, rather than the arithmetic mean.
Hard
55.Explain why the median is often a better average than the mean when a data set contains an outlier.
Hard
56.Calculate the mean of the following data set: 2, 3, 4, 7, 9.
Medium
- A7
- B4
- C6
- D5
57.The mean of five numbers is 10. Four of the numbers are 8, 9, 11, and 12. What is the fifth number?
Medium
- A12
- B11
- C10
- D9
58.Find the mean of the data set: 3, 5, 7, 9, 11.
Medium
- A9
- B7
- C6
- D8
59.The mean of five numbers is 12. Four of the numbers are 10, 14, 9, and 15. Find the fifth number.
Medium
- A14
- B12
- C13
- D11
60.A data set has a mean of 15 and a range of 6. Every value is multiplied by 2. What is the new mean?
Medium
- A33
- B30
- C27
- D36
61.The mean of five numbers is 12. Four of the numbers are 9, 11, 14 and 16. Find the fifth number.
Medium
- A9
- B12
- C11
- D10
62.A data set contains the values 4, 7, 7, 9, 13, 18. Find the range of the data set.
Medium
- A14
- B16
- C15
- D13

Resumos de revisão

## Averages and Range — KS3 Revision **Average** = a value that is most central, common, or typical of a data set. The three main averages are the **mean**, **median** and **mode** ("the three Ms"). ### The three averages - **Mean** (arithmetic mean) = sum of values ÷ how many values there are. - Example: for 2, 3, 4, 7, 9 the mean is (2+3+4+7+9)/5 = 25/5 = 5. - **Median** = the middle value once the data is sorted in order. - Odd number of values: the single middle one. - Even number of values: the mean of the two middle values. - **Mode** = the most common value (the one that appears most often). - Useful for categorical (non-numeric) data, and when you want the value representing the majority. ### Range - **Range** = largest value − smallest value. It measures how spread out the data is (dispersion), not the centre. ### Choosing the best average - Use the **median** when data is **skewed** or has **outliers**, so the long tail doesn't distort the picture. Example: average personal income is usually given as the median income, so it represents most people rather than being pulled up by a few very high earners. - Use the **mode** for categorical data or when the most frequent value matters. - Use the **mean** when the data is fairly even and you want to use every value. ### Missing value given the mean If you know the mean and all but one value: - total of all values = mean × number of values - missing value = total − sum of the known values ### Frequency tables - **Mean from a frequency table**: multiply each value by its frequency, add these up, then divide by the total frequency. - **Modal class** = the group (class interval) with the highest frequency. - **Estimated mean from grouped data**: use the **midpoint** of each class as the value, multiply by frequency, add up, divide by total frequency. It is an *estimate* because exact values inside each group are unknown. ### Comparing two data sets Compare using **one average** (mean, median or mode) **and the range**: - the average tells you which set is typically higher/lower; - the range tells you which set is more spread out/consistent. ### Key properties of averages - Every average lies between the minimum and maximum of the data. - If all values are the same, every average equals that value. - Multiplying every value by the same scale factor multiplies the average by that factor. - The order of the list does not change the average. ### Notation - Sample mean is written \(\bar{x}\) (x-bar); the mean of a whole population/distribution is written \(\mu\).

Average: A value that is most central, common, or typical of a collection of data, representing its overall position. Most commonly the arithmetic mean, but also the median or mode.
Mean (arithmetic mean): The sum of the values divided by the count of values. For 2, 3, 4, 7, 9: (2+3+4+7+9)/5 = 5.
Median: The value in the centre after sorting the data into order. Used when data is skewed or has outliers.
Mode: The most common value in the group. Can be used for categorical data or when frequency matters.
Range: The largest value minus the smallest value; a measure of how spread out the data is.
Central tendency: A central or typical value for a set of data or distribution; informally called an average. The most common measures are the mean, median and mode.
Dispersion (variability): How spread out the data is, contrasted with central tendency. The range is a simple measure of dispersion.
Outlier: A value far from the rest of the data; it can pull the mean away from the typical value, which is why the median is often preferred.
Modal class: In a frequency table of grouped data, the class interval with the highest frequency.
Estimated mean from grouped data: An estimate of the mean found by using the midpoint of each class interval as the value, multiplying by its frequency, adding, and dividing by the total frequency.
Sample mean: The arithmetic mean of a sample, written \(\bar{x}\), as distinct from the mean of the whole underlying group.
Population mean: The mean of the whole underlying distribution, denoted \(\mu\) (or \(\mu_x\)).

Flashcards

What is the arithmetic mean of a list of numbers?

The arithmetic mean is the sum of all the values divided by the number of values. It is the most commonly used average.

How do you find the median of a list of numbers?

Sort the numbers into order, then find the middle value. If there is an even number of values, the median is the mean of the two middle values.

What is the mode of a data set?

The mode is the value that occurs most often. A data set can have no mode, one mode, or more than one mode.

How is the range of a data set calculated?

Range = largest value − smallest value.

Why might the median be a better average than the mean for data with outliers?

The median is not affected by extreme values, so it better represents the typical value when the data is skewed or has outliers.

When is the mode the most appropriate average to use?

The mode is best for categorical data or when you want the most common or popular value.

How do you find a missing value when the mean of a data set is known?

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 value.

How do you calculate the mean from a frequency table?

Multiply each value by its frequency, add these products to get the total, then divide by the total frequency.

How do you estimate the mean from grouped data?

Use the midpoint of each class interval as the value, multiply each midpoint by its frequency, sum these products, then divide by the total frequency.

What is the modal class?

The modal class is the class interval with the highest frequency in grouped data.

How do you compare two data sets using an average and the range?

Compare a typical value (mean or median) to describe the overall level, and compare the range to describe the spread or consistency.

What does it mean if a data set has a small range?

A small range means the data values are close together, so the data is more consistent.

What does it mean if a data set has a large range?

A large range means the data values are spread out, so the data is more variable.

What is a measure of central tendency?

A measure of central tendency is a single value that describes the centre or typical value of a data set. The mean, median and mode are examples.

What is the difference between the mean and the median?

The mean uses all values and can be affected by extremes; the median depends only on the middle value(s) after sorting and is not affected by extremes.

If every value in a data set is the same, what are the mean, median and mode?

They are all equal to that same value.

What happens to the mean if every value in a data set is multiplied by 3?

The mean is also multiplied by 3.

Does changing the order of the values in a list change the mean, median or mode?

No, the mean, median and mode are not affected by the order of the values.

What is the harmonic mean used for?

The harmonic mean is used for rates or ratios, such as average speed over the same distance.

Why is median income often used instead of mean income?

Median income better represents the typical person because it is not influenced by the very high incomes of a few people.

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