Statistics — Grade X

Grade X

STATISTICS || तथ्याङ्कशास्त्र

Statistics तथ्याङ्कशास्त्र

Math — Grade X · Chapter 6

📚 5 Topics

🎯 TQ 8 · TM 11

🔢 K:2 · U:2 · A:2 · HA:2

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Statistics

Click any topic to expand subtopics

Exercise — Model 1

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Key Formulas

Statistics — essential formulas

Arithmetic Mean — Individual

$$\bar{x} = \frac{\sum x}{n}$$

Sum of all values divided by the number of values $n$.

Arithmetic Mean — Frequency

$$\bar{x} = \frac{\sum fx}{\sum f}$$

$f$ = frequency, $x$ = midpoint of class interval. Used for grouped data.

Mean — Short-Cut Method

$$\bar{x} = A + \frac{\sum fd}{\sum f}$$

$A$ = assumed mean, $d = x - A$ = deviation. Simplifies calculation.

Median — Individual / Discrete

$$M = \left(\frac{n+1}{2}\right)^{\text{th}} \text{ value}$$

Arrange data in order first. For even $n$, average the two middle values.

Median — Continuous (grouped)

$$M = L + \frac{\frac{n}{2} - cf}{f} \times h$$

$L$ = lower boundary of median class, $cf$ = cumulative freq. before, $f$ = freq. of median class, $h$ = class width.

Lower Quartile Q₁

$$Q_1 = L + \frac{\frac{n}{4} - cf}{f} \times h$$

Locate the $\frac{n}{4}$th value. $L$, $cf$, $f$, $h$ follow the same median-class logic.

Upper Quartile Q₃

$$Q_3 = L + \frac{\frac{3n}{4} - cf}{f} \times h$$

Locate the $\frac{3n}{4}$th value. Used to find the interquartile range.

Interquartile Range

$$IQR = Q_3 - Q_1$$

Measures the spread of the middle 50% of data. Not affected by extreme values.

Mode — Grouped Data

$$Mo = L + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h$$

$f_1$ = modal class freq., $f_0$ = freq. before, $f_2$ = freq. after, $h$ = class width.

Quartile Deviation

$$QD = \frac{Q_3 - Q_1}{2}$$

Also called semi-interquartile range. A measure of dispersion.

Coefficient of Quartile Deviation

$$CQD = \frac{Q_3 - Q_1}{Q_3 + Q_1}$$

A relative measure useful for comparing two distributions.

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Key Concepts

Understand beyond memorising formulas

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Arithmetic Mean

The mean is the "balance point" of a dataset. For grouped data use $\bar{x}=\Sigma fx/\Sigma f$ with midpoints. The short-cut method with an assumed mean reduces arithmetic errors without changing the result.

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Median

The median is the middle value when data is ordered — it splits data into two equal halves. For grouped data, find the median class where cumulative frequency first reaches or exceeds $n/2$, then apply the interpolation formula.

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Quartiles

Quartiles split data into four equal parts. Q₁ is the lower quartile (25th percentile), Q₂ is the median, and Q₃ is the upper quartile (75th percentile). The IQR = Q₃ − Q₁ measures the middle 50% spread.

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Mode

The mode is the most frequently occurring value. A dataset can have no mode, one mode (unimodal), or more (bimodal/multimodal). For grouped data, the modal class has the highest frequency; use the formula to estimate the exact mode.

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Cumulative Frequency

Build a cumulative frequency table by adding each class frequency to the running total. Plot it as an ogive (cf curve). Read off median, Q₁, and Q₃ graphically from the ogive at n/2, n/4, and 3n/4.

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Choosing the Right Average

Use mean when data is symmetric and has no extreme values. Use median when there are outliers or skewed data (e.g. income). Use mode for categorical data or when the most common value matters (e.g. most popular shoe size).

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Class Interval & Midpoint

For grouped data, the midpoint (class mark) = (lower + upper boundary) ÷ 2. Always use true/actual class boundaries (not stated limits) for continuous data when computing median, quartiles, and mode.

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