Averages and spread

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Apuntes de la lección Diapositivas Preguntas de práctica

Apuntes de la lección

Measures of Central Tendency

Mean = sum of all data values ÷ number of values. It uses every value, so it is sensitive to outliers.
Median = middle value when data are ordered from smallest to largest. For an even number of values, it is the mean of the two middle values.
Mode = most frequently occurring value. A data set can have more than one mode or no mode.
Modal class = the class interval with the highest frequency in a grouped frequency table.
Range = largest value − smallest value. It measures spread but can be distorted by outliers.

Averages from a List

To find the mean: add all values, then divide by the number of values.
To find the median: first order the data, then locate the middle value (or average the two middle values if n is even).
To find the mode: identify the value that appears most often.
To find the range: subtract the smallest value from the largest.

Averages from a Frequency Table

For a frequency table with values x and frequencies f, the mean is \frac{\sum (x \times f)}{\sum f}.
The total frequency \sum f is the number of data values, not the number of rows.
The median is the value at position \frac{n+1}{2} (or the average of the two middle values if n is even) in the ordered data.
The mode is the value with the highest frequency.
The range is the difference between the largest and smallest values in the table.

Mean from frequencies

Grouped Frequency Tables

For grouped data, use the midpoint of each class as a representative value.
To estimate the mean: multiply each midpoint by its frequency, sum these products, then divide by the total frequency.
The estimated mean is an approximation, not an exact value.
The modal class is the class with the highest frequency.
The median class is the class that contains the middle value; find it by cumulative frequency.

Estimating a grouped mean

Quartiles and Interquartile Range

Quartiles divide ordered data into four equal parts: lower quartile (Q1), median (Q2), upper quartile (Q3).
Lower quartile (Q1) = median of the lower half of the data.
Upper quartile (Q3) = median of the upper half of the data.
Interquartile range (IQR) = Q3 − Q1. It measures the spread of the middle 50% of the data.
IQR is generally less sensitive to outliers than the full range. Adding or removing observations can still change quartiles.
Worked list examples use medians of the halves, excluding the overall median when n is odd. Follow the convention specified by the question or course.

Quartiles and interquartile range

Choosing the Most Appropriate Average

Use the mean when data are symmetric and have no outliers; it uses all values.
Use the median when there are outliers or skewed data; it is resistant to extreme values.
Use the mode for categorical data or when the most common value is needed.
Justify your choice by referring to the context and the effect of outliers.

Comparing Distributions

Compare two data sets using one measure of central tendency (mean or median) and one measure of spread (range or IQR).
If outliers are present, use the median and IQR for comparison.
If data are symmetric and outlier-free, the mean and range may be appropriate.
Always state which average and which spread you are using and explain what they show about the data.

Common Misconceptions

When finding the mean from a frequency table, divide by the total frequency, not the number of classes.
The median is the middle value, not the middle row of the table.
Always order the data before finding the median.
The mode is the value with the highest frequency, not the frequency itself.
An estimated mean from grouped data is not exact; it is an approximation.
The range can be distorted by outliers; consider the IQR instead.

Diapositivas

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Preguntas de práctica

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1.What is the mode of a data set?
Easy
- AThe middle value when the data are ordered
- BThe value that occurs most often
- CThe arithmetic average of all values
- DThe difference between the largest and smallest values
2.Which measure of spread is calculated by subtracting the smallest value from the largest value?
Easy
- AInterquartile range
- BVariance
- CRange
- DStandard deviation
3.To find the median, order the data: use the central value when the count is odd, or the mean of the two central values when the count is even.
Easy
True or false?
4.The heights (in cm) of five students are: 150, 155, 160, 165, 170. What is the median height?
Medium
- A155 cm
- B160 cm
- C162.5 cm
- D165 cm
5.The ages of people in a queue are: 12, 15, 15, 17, 19, 21, 24. What is the interquartile range? Use medians of the lower and upper halves, excluding the overall median when the count is odd.
Medium
- A6
- B7
- C9
- D12
6.Households have 0, 1, 2, 3 and 4 pets with frequencies 3, 5, 4, 2 and 1 respectively. Find the mean number of pets to 1 decimal place.
Medium
- A1.2
- B1.4
- C1.5
- D2.0
7.Which of the following statements about the mean and median are true? (Select all that apply)
Medium
- AThe mean is affected by extreme values (outliers).
- BThe median is always equal to the mean.
- CThe median is generally less sensitive to extreme values than the mean.
- DThe mean is always the most appropriate average.
- EThe median is the middle value when data are ordered.
8.When estimating the mean from a grouped frequency table, you should divide the total of (midpoint × frequency) by the number of classes.
Medium
True or false?

Más preguntas

9.A data set has a mean of 20 and a range of 10. If each value is multiplied by 3, what are the new mean and range?
Hard
- AMean 60, range 30
- BMean 20, range 10
- CMean 60, range 10
- DMean 20, range 30
10.Arrange the following steps in the correct order to find the median of a data set.
Easy
- Order the data from smallest to largest.
- If there is an even number of values, average the two middle values.
- Locate the middle value (or values).
11.Match each measure to its correct definition.
Medium
- Mean
- Median
- Mode
- The middle value when data are ordered
- The value that occurs most often
- The sum of values divided by the number of values
12.For the grouped data: 0-4 (frequency 3), 5-9 (frequency 7), 10-14 (frequency 10), 15-19 (frequency 5), what is the estimated mean? (Use midpoints.)
Hard
- A9.2
- B9.6
- C10.0
- D10.4
13.Match each measure of central tendency to its correct definition.
Easy
- Mean
- Median
- Mode
- The middle value when data are ordered
- The most frequent value
- The sum of values divided by the number of values
14.Arrange the steps in the correct order to find the median of a data set.
Medium
- Count the number of values
- Order the data from smallest to largest
- Identify the middle value (or average the two middle values)
15.Which of the following are correct statements about the interquartile range (IQR)? (Select all that apply)
Medium
- AIt is the range of the middle 50% of the data
- BIt is calculated as Q3 - Q1
- CIt is always larger than the range
- DIt is less affected by outliers than the range
- EIt is the same as the standard deviation
16.The mode can only be used for numerical data.
Easy
True or false?
17.When estimating the mean from a grouped frequency table, you divide the total of (midpoint × frequency) by the number of classes.
Easy
True or false?
18.The mean of a data set is 20 and the range is 10. If each value is multiplied by 3, what is the new range?
Medium
- A10
- B30
- C60
- D90
19.A data set has five values: 4, 8, 12, 16, and 20. Which measure of spread is most affected by an outlier if one is added?
Hard
- ARange
- BInterquartile range
- CMedian
- DMode
20.The heights (in cm) of 7 plants are: 12, 15, 10, 18, 15, 20, 14. What is the mode?
Medium
- A20
- B14
- C15 cm, because it appears twice
- D14.86
21.For the equal-width classes 0 ≤ t < 5, 5 ≤ t < 10, 10 ≤ t < 15 and 15 ≤ t < 20 minutes, frequencies are 4, 9, 12 and 5. What is the modal class?
Medium
- A10-15
- B12
- C5-10
- D15-20
22.The ages of 9 people are: 12, 15, 15, 17, 19, 21, 24, 28, 35. What is the median age?
Medium
- A19
- B17
- C21
- D18
23.For the data set: 5, 7, 8, 10, 12, 15, 18, 20. What is the interquartile range (IQR)? Use medians of the lower and upper halves, excluding the overall median when the count is odd.
Medium
- A8
- B10
- C9
- D13
24.The range is calculated by subtracting the smallest value from the largest value.
Easy
True or false?
25.Which of the following statements about the mean, median, and mode are correct? (Select all that apply)
Medium
- AThe mean is affected by outliers.
- BThe median is always a value from the data set.
- CThe mode can be used for non-numeric data.
- DThe range is a measure of central tendency.
- EThe interquartile range is a measure of spread.
26.The prices (in £) of 8 books are: 5, 7, 8, 10, 12, 15, 18, 20. Which measure of spread is most affected if a new book costing £100 is added?
Medium
- ARange
- BInterquartile range
- CMode
- DMedian
27.A data set has values: 3, 7, 8, 10, 12, 15, 18, 20, 22, 25. Which of the following statements are correct? (Select all that apply) Use medians of the lower and upper halves, excluding the overall median when the count is odd.
Medium
- AThe median is 13.5
- BThe lower quartile is 8
- CThe upper quartile is 18
- DThe interquartile range is 12
- EThe range is 25
28.
When estimating the mean from a grouped frequency table, you divide the total of (midpoint × frequency) by the total frequency.
Midpoint (x) Frequency (f) x × f
5 2 10
15 3 45
25 4 100
35 1 35
Total 10 190
Easy
True or false?
29.The heights (in cm) of 10 plants are: 10, 12, 14, 15, 15, 16, 18, 20, 22, 25. Which measure of central tendency is most appropriate to represent the typical height if there is an outlier of 100 cm added?
Medium
- AMean
- BMedian
- CMode
- DRange
30.Arrange the following steps in the correct order to find the interquartile range of a data set. Use medians of the lower and upper halves, excluding the overall median when the count is odd.
Medium
- Find the median of the data set
- Order the data from smallest to largest
- Find the median of the lower half (Q1)
- Find the median of the upper half (Q3)
- Subtract Q1 from Q3
31.Which of the following is a measure of central tendency?
Easy
- ARange
- BMean
- CInterquartile range
- DVariance
32.Which of the following is a measure of spread?
Easy
- AMode
- BMedian
- CRange
- DMean
33.The mode is the value that appears most often in a data set.
Easy
True or false?
34.The interquartile range is calculated by subtracting the smallest value from the largest value.
Easy
True or false?
35.For the data set: 3, 7, 7, 9, 12, 15, 18, 22, what is the median?
Medium
- A9
- B10.5
- C7
- D12
36.The mean of four numbers is 15. Three of the numbers are 12, 18, and 10. What is the fourth number?
Medium
- A20
- B15
- C25
- D16
37.Which of the following statements about the range are correct? (Select all that apply)
Medium
- AThe range is a measure of spread.
- BThe range is calculated by subtracting the smallest value from the largest value.
- CThe range is always a positive number or zero.
- DThe range is the most appropriate measure of spread when there are outliers.
- EThe range uses all data values in its calculation.
38.
When estimating the mean from a grouped frequency table, you should divide the total of (midpoint × frequency) by the total frequency.
Class interval Midpoint Frequency
0-10 5 4
10-20 15 6
20-30 25 10
30-40 35 5
40-50 45 3
50-60 55 2
Easy
True or false?
39.Match each statistical term to its definition.
Medium
- Range
- Interquartile range
- Quartiles
- The difference between the largest and smallest values
- The difference between the upper quartile and the lower quartile
- Values that divide the data into four equal parts
40.The heights (in cm) of 8 students are: 150, 155, 160, 162, 165, 168, 170, 175. What is the median height?
Medium
- A162 cm
- B163.5 cm
- C165 cm
- D160 cm
41.Complete the sentence with the correct statistical term.
Easy
The difference between the upper quartile and the lower quartile is called the ____.
42.Find the mode of the following data set: 3, 5, 7, 5, 9, 5, 2.
Easy
43.State the formula for the mean of a data set in terms of the sum of values and the number of values.
Easy
Mean = ____ / ____
44.Calculate the mean of the following numbers: 4, 8, 6, 10, 12. Give your answer as a decimal.
Medium
45.The mean of five numbers is 12. Four of the numbers are 8, 10, 14, and 16. Find the fifth number.
Medium
46.State the modal class for the grouped frequency table: 0-9 (frequency 5), 10-19 (frequency 8), 20-29 (frequency 12), 30-39 (frequency 7).
Medium
47.Complete the sentence about estimating a median from a cumulative-frequency graph.
Medium
For a cumulative-frequency graph with total frequency n, read the estimated median at cumulative frequency ____.
48.The lower quartile of a data set is 15 and the upper quartile is 27. What is the interquartile range?
Hard
49.The student heights in cm are 150, 155, 160, 160, 165, 170, 175, 180, 185, 190. Find the mean height.
Medium
50.The student heights in cm are 150, 155, 160, 160, 165, 170, 175, 180, 185, 190. Find the median height.
Medium
51.The student heights in cm are 150, 155, 160, 160, 165, 170, 175, 180, 185, 190. Find the range of the heights.
Medium
52.For the frequency table below, calculate the mean number of books read per student. Give your answer to 1 decimal place. Books read (x): 0, 1, 2, 3, 4. Frequency (f): 3, 5, 6, 4, 2.
Medium
53.Explain why the median might be a more appropriate measure of central tendency than the mean when comparing two data sets that contain outliers.
Hard
54.Two classes took the same test. Class A had a mean score of 65 and a range of 40. Class B had a mean score of 65 and a range of 10. Compare the performance of the two classes, using the mean and the range.
Hard
55.Two classes took the same test. Class A had a mean score of 65 and a range of 40. Class B had a mean score of 65 and a range of 10. Compare the performance of the two classes.
Hard
56.Find the mode of the following data set: 3, 5, 7, 5, 9, 5, 2.
Easy
- A6
- B5
- C7
- D4
57.Calculate the mean of the following numbers: 4, 8, 6, 10, 12. Give your answer as a decimal.
Medium
- A10
- B7
- C9
- D8
58.The mean of five numbers is 12. Four of the numbers are 8, 10, 14, and 16. Find the fifth number.
Medium
- A13
- B14
- C11
- D12
59.The lower quartile of a data set is 15 and the upper quartile is 27. What is the interquartile range?
Hard
- A13
- B11
- C14
- D12
60.The student heights in cm are 150, 155, 160, 160, 165, 170, 175, 180, 185, 190. Find the mean height.
Medium
- A186 cm
- B152 cm
- C203 cm
- D169 cm
61.The student heights in cm are 150, 155, 160, 160, 165, 170, 175, 180, 185, 190. Find the median height.
Medium
- A201.1 cm
- B150.7 cm
- C167.5 cm
- D184.3 cm
62.The student heights in cm are 150, 155, 160, 160, 165, 170, 175, 180, 185, 190. Find the range of the heights.
Medium
- A40 cm
- B44 cm
- C36 cm
- D48 cm
63.For the frequency table below, calculate the mean number of books read per student. Give your answer to 1 decimal place. Books read (x): 0, 1, 2, 3, 4. Frequency (f): 3, 5, 6, 4, 2.
Medium
- A1.9 books
- B2.8 books
- C3.8 books
- D0.8 books

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Midpoint (x)	Frequency (f)	x × f
5	2	10
15	3	45
25	4	100
35	1	35
Total	10	190

Class interval	Midpoint	Frequency
0-10	5	4
10-20	15	6
20-30	25	10
30-40	35	5
40-50	45	3
50-60	55	2