MathematicsChapter 11: Basic Statistics
The Mean
An average locates the data with one number.
Learning objectives
- Explain the mean in clear mathematical language.
- Use the main rule, formula, test, or construction correctly.
- Work through a simple example and show each important step.
- Identify common mistakes and write an exam-ready conclusion.
Definition
An average locates the data with one number. The mean is the balancing point. The unit symbol is x-bar (read 'x bar') for the mean. For ungrouped data, x-bar = sum xn. If values come with frequencies, x-bar = sum f xn, where n = Sigma f. For grouped data you do not have each x, so you use class marks as the x in that formula. A useful rewrite is n * mean = sum, which lets you find a missing total or a missing value.
Mathematical form
bar = sum xn. If values come with frequenci
bar = sum f xn
n = Sigma f. For grouped data you do not hav
mean = sum
Where / symbols
- mean
- The mean is the balancing point.
- x-bar
- The unit symbol is x-bar (read 'x bar') for the mean.
- (sum x)/n
- For ungrouped data, x-bar = (sum x)/n.
- (sum f x)/n
- If values come with frequencies, x-bar = (sum f x)/n, where n = Sigma f.
- class marks
- For grouped data you do not have each x, so you use class marks as the x in that formula.
- n * mean = sum
- A useful rewrite is n * mean = sum, which lets you find a missing total or a missing value.
Relationship
- n * mean = sum
Overview
Same idea, three levels of formula:
Ungrouped and Frequency Mean
Ungrouped: add every number and divide by how many numbers you added. Frequency table of single values (discrete): multiply each value by its frequency, add those products, divide by Sigma f. A value that appears often pulls x-bar toward itself more than a rare value does.
Examples 2, 4, 6, 8: x-bar = 20/4 = 5 · Values 1, 2, 3 with f = 2, 3, 5: sum f x = 2+6+15 = 23, n = 10, x-bar = 2.3 · If n * mean = 50 and n = 10, the total sum is 50
Grouped Mean
Replace each class by its class mark x, form f x, then x-bar = sum f xn. This assumes the values in a class sit at the middle. That is why a grouped mean is an estimate. Do not use limits or boundaries in the x column; use marks. Check Sigma f = n before you divide.
Examples Class 1-10, f = 4, mark 5.5, contribution 4*5.5 = 22 · Add all f x, divide by Sigma f · Changing h changes the marks, so recompute; do not reuse an old total
WHAT — Core idea
An average locates the data with one number. The mean is the balancing point. The unit symbol is x-bar (read 'x bar') for the mean. For ungrouped data, x-bar = sum xn. If values come with frequencies, x-bar = sum f xn, where n = Sigma f. For grouped data you do not have each x, so you use class marks as the x in that formula. A useful rewrite is n * mean = sum, which lets you find a missing total or a missing value.
HOW — Rules, method, and examples
1. UNGROUPED AND FREQUENCY MEAN: Ungrouped: add every number and divide by how many numbers you added. Frequency table of single values (discrete): multiply each value by its frequency, add those products, divide by Sigma f. A value that appears often pulls x-bar toward itself more than a rare value does. 2. GROUPED MEAN: Replace each class by its class mark x, form f x, then x-bar = sum f xn. This assumes the values in a class sit at the middle. That is why a grouped mean is an estimate. Do not use limits or boundaries in the x column; use marks. Check Sigma f = n before you divide.
WHY — Meaning and connections
Mean uses every value, so an extreme outlier pulls x-bar hard. That is a reason the next page also teaches median and mode. If a question gives the mean and all but one value, use n * mean = sum to recover the missing number.
HOW — Answer in an exam
Write x-bar for the mean. Ungrouped: sum xn. With frequencies: sum f xn. Grouped: x is the class mark. Identity: n * mean = sum. Last check: Sigma f = n before you divide.
Explanation
Mean uses every value, so an extreme outlier pulls x-bar hard. That is a reason the next page also teaches median and mode. If a question gives the mean and all but one value, use n * mean = sum to recover the missing number.
In simple words
Write x-bar for the mean. Ungrouped: sum xn. With frequencies: sum f xn. Grouped: x is the class mark. Identity: n * mean = sum. Last check: Sigma f = n before you divide.
Key points
- mean
- x-bar
- sum xn
- sum f xn
- add every number
- multiply each value by its frequency
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.The Mean
Sticky Notes · Quick Revision
WHAT
- An average locates the data with one number.
Key Points
- An average locates the data with one number.
- Ungrouped: add every number and divide by how many numbers you added.
- Replace each class by its class mark x, form f x, then x-bar = sum f xn.
- Mean uses every value, so an extreme outlier pulls x-bar hard.
HOW
- 1.
- Ungrouped: add every number and divide by how many numbers you added.
- Replace each class by its class mark x, form f x, then x-bar = sum f xn.
Examples
- 2, 4, 6, 8: x-bar = 204 = 5
- Values 1, 2, 3 with f = 2, 3, 5: sum f x = 2+6+15 = 23, n = 10, x-bar = 2.3
- If n * mean = 50 and n = 10, the total sum is 50
- Class 1-10, f = 4, mark 5.5, contribution 4*5.5 = 22
- Add all f x, divide by Sigma f
Common Mistakes
- Correction: Write x-bar for the mean.
- Correction: Ungrouped: sum xn.
- Correction: With frequencies: sum f xn.
Exam Tip
- Write x-bar for the mean. Ungrouped: sum xn. With frequencies: sum f xn. Grouped: x is the class mark. Identity: n * mean = sum. Last check: Sigma f = n before you divide.
60-second revision
- An average locates the data with one number.
- Ungrouped: add every number and divide by how many numbers you added.
- Replace each class by its class mark x, form f x, then x-bar = sum f xn.
- Mean uses every value, so an extreme outlier pulls x-bar hard.
- Write x-bar for the mean.
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