grade 9 Representative Values & Spread

Mean, Median, Mode & Standard Deviation

A test drops 30 scores at once, and one representative value sums them into a single number. Besides the mean, the median is the middle value when sorted and the mode is 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

Spread1.5

👀 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? The first graph has 20 points around 70. The spread mark 1.5 is a relative scale, not sample σ. The 5-person card 60, 70, 70, 80, 100 is a different set. The variance formula uses /n

Three Representative Values

Mean

x̄ = x₁ + x₂ + ... + x_nn

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

5th student's score80

⚠️ 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)²_nn

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.

Arrange the data in order (already sorted).

1, 3, 5, 7, 9

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.

Find the mean: (1+3+5)/3 = 3.

mean = (1+3+5)/3 = 3

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

Mean = (sum) ÷ (count), so the sum = 7 × 5 = 35.

5+6+8+9+x = 35

28 + x = 35, so x = 7.

28+x=35 ⇒ x=7

🎯 Exam Points

①Sum of deviations = 0 (always — useful for verification)

②Variance = mean of squared deviations, σ = √V

③Small σ → data is tightly grouped

④Outliers present → use median, not mean

⑤Variance unit = (original)², σ unit = original

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Inscribed Angle

Correlation

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