A test hands you 30 scores, and one representative value sums them all up.
The mean is the average, the median the middle value when sorted, the mode the most frequent.
Spread shows how far data strays from the mean via variance and its root, the standard deviation.
Slide the spread and the outlier: an outlier jolts the mean but hardly moves the median.
Intuition — Summarize a dataset with one number
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Data on a number line — change the standard deviation to observe the spread
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👀 Why we need a representative value
①A test gives 30 student scores
②You can't memorize 30 numbers
③Summarize with a single number → that's the representative value
④Mean, median, mode — when do we use which?
Three Representative Values
Mean
x̄ = x1 + x2 + ... + xnn
Sum of all values divided by count. The most-used representative value
Median
Sort data → middle value
For even n, average of two middle values. Less affected by outliers
Mode
The most frequent value
Can be multiple. Best for categorical data (color, gender)
💡 Concrete Example
①Five scores: 60, 70, 70, 80, 100
②Mean = (60+70+70+80+100)/5 = 76
③Median = 3rd value = 70 (middle when sorted)
④Mode = 70 (appears twice — most often)
⑤Mean (76) ≠ median (70) — the 100 lifted the mean!
Outliers & Choosing the Right Value
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How an outlier affects the mean — raise the 5th student's score
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⚠️ With an Outlier
①Extremely large/small value = outlier
②An outlier skews the mean significantly
③But the median barely changes
④e.g. average salary is high due to a few top earners → median is more realistic
⑤Exam: 'best value when outliers exist?' → median!
Spread — Deviation, Variance, Std. Dev.
Deviation
deviation = (value) - (mean)
How far each value is from the mean. Sum of deviations is always 0!
Variance
V = (dev)21 + (dev)22 + ... + (dev)2nn
Mean of squared deviations. We square because the sum of deviations is 0
Standard Deviation
σ = √V = √(variance)
Square-rooted variance — restored to original units
🔍 Why Square?
①Sum of deviations cancels out to 0
②Square them so all are positive, then average
③This is variance V
④Variance has unit of (score)² which is hard to interpret
⑤Take square root → back to original unit → standard deviation
Work It Out
Example 1
Find the median of the data 1, 3, 5, 7, 9.
1
Arrange the data in order (already sorted).
1, 3, 5, 7, 9
2
With 5 values (odd), the median is the 3rd (middle) value.
median = 5
▸ 5
Odd count → middle value; even count → average of the two middle values.
Example 2
Find the variance of the data 1, 3, 5.
1
Find the mean: (1+3+5)/3 = 3.
mean = (1+3+5)/3 = 3
2
Average the squared deviations: ((−2)²+0²+2²)/3.
variance = (4+0+4)/3 = 8/3
▸ 8/3
Variance = mean of squared deviations. Deviation = value − mean.
Exam Wrap-up
Spread Core
V = Σ(dev)2n, σ = √V
Smaller σ → data clusters tightly around the mean
G9 school-exam type
The mean of the five values 5, 6, 8, 9, x is 7. Find x.
①5
②6
③7
④8
⑤9
▸ ③ 7
1
Mean = (sum) ÷ (count), so the sum = 7 × 5 = 35.
5+6+8+9+x = 35
2
28 + x = 35, so x = 7.
28+x=35 ⇒ x=7
🎯 Exam Points
①Sum of deviations = 0 (always — useful for verification)