Chapter 13 — Statistics: Quality Revision Notes

These notes are fully paraphrased and copyright-conscious. They focus on concepts, formulas, method selection, and exam-useful points rather than reproducing the textbook. The chapter extends mean, median and mode from ungrouped to grouped data and introduces cumulative frequency.

1. Core Idea: Measures of Central Tendency

A measure of central tendency gives a representative value for a data set.

Measure	What it tells us	Best used when
Mean	Overall average	All observations matter
Median	Middle/typical value	Extreme values may distort the mean
Mode	Most frequent value	Most popular/common value is required

The choice depends on what the data is being used to describe.

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2. Mean of Grouped Data

For a frequency distribution, let:

• $x_i$ = observation/class mark
• $f_i$ = corresponding frequency
• $\sum f_i$ = total frequency

Direct Method

$$\boxed{\bar{x}=\frac{\sum f_i x_i}{\sum f_i}}$$

For grouped data, each class is represented by its class mark:$$\boxed{x_i=\frac{\text{lower limit}+\text{upper limit}}{2}}$$

The textbook treats the class mark as representative of the observations in that class. Therefore, the resulting mean for grouped data is generally an approximation when the original observations are unknown.

Example idea

For class $40-55$:$$x_i=\frac{40+55}{2}=47.5$$

Then multiply this class mark by its frequency and use the direct formula.

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3. Assumed Mean Method

When the $x_i$ and $f_i$ values are large, direct multiplication can become lengthy.

Choose a convenient value $a$, called the assumed mean, usually a class mark near the centre.

Define deviation:$$\boxed{d_i=x_i-a}$$

Then:$$\boxed{\bar{x}=a+\frac{\sum f_i d_i}{\sum f_i}}$$

Steps

1. Find all class marks $x_i$.
2. Select a convenient assumed mean $a$.
3. Calculate $d_i=x_i-a$.
4. Calculate $f_i d_i$.
5. Find $\sum f_i$ and $\sum f_i d_i$.
6. Substitute in the formula.

An important point from the chapter: the final mean does not depend on which class mark is selected as the assumed mean.

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4. Step-Deviation Method

If the deviations have a common factor, calculations can be shortened further.

Define:$$\boxed{u_i=\frac{x_i-a}{h}}$$

where:

• $a$ = assumed mean
• $h$ = suitable common factor, usually class size

Then:$$\boxed{\bar{x}=a+h\left(\frac{\sum f_i u_i}{\sum f_i}\right)}$$

When is it useful?

Use it particularly when:

• class intervals have a common size, and
• the deviations are convenient multiples of a common number.

The chapter notes that step-deviation is essentially a simplified version of the assumed-mean method.

Quick comparison

$$\boxed{\text{Direct: }\bar{x}=\frac{\sum fx}{\sum f}}$$$$\boxed{\text{Assumed mean: }\bar{x}=a+\frac{\sum fd}{\sum f}}$$$$\boxed{\text{Step deviation: }\bar{x}=a+h\frac{\sum fu}{\sum f}}$$

All three methods give the same grouped-data mean when applied to the same grouped distribution.

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5. How to Choose the Mean Method

Direct Method

Prefer when the numbers involved are relatively small.

Assumed Mean Method

Useful when the class marks are large and subtracting a convenient value makes calculations easier.

Step-Deviation Method

Best when the deviations have a common factor, making $u_i$ small and simple.

The chapter specifically recommends choosing the method according to the numerical size and structure of the data.

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6. Mode of Grouped Data

The mode is the value occurring most frequently.

For grouped data, we first identify the class with the highest frequency.

This is called the:$$\boxed{\text{Modal class}}$$

For equal-width continuous classes:$$\boxed{ \text{Mode} = l+ \frac{f_1-f_0}{2f_1-f_0-f_2}\times h }$$

where:

• $l$ = lower limit of modal class
• $h$ = class width
• $f_1$ = frequency of modal class
• $f_0$ = frequency immediately before modal class
• $f_2$ = frequency immediately after modal class

Exam method

Step 1: Find the highest frequency.
Step 2: Identify its class → modal class.
Step 3: Write $l,h,f_0,f_1,f_2$.
Step 4: Substitute into the formula.

The chapter restricts its treatment to problems with a single mode, although grouped data can theoretically have more than one modal value.

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7. Mean vs Mode

These two measures answer different questions.

• Mean: What is the overall average?
• Mode: What value is most common?

For example, in the chapter’s marks distribution, the grouped-data mode is 52 while the mean is 62. This means the most concentrated/typical frequency occurs around the modal value, whereas the mean reflects the overall average.

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8. Cumulative Frequency

Cumulative frequency (cf) is obtained by progressively adding frequencies.

Example:

Class	Frequency	Cumulative frequency
First	5	5
Second	3	8
Third	4	12
Fourth	3	15

So the cumulative frequency tells us how many observations have accumulated up to that point.

Less-than type

Counts observations below successive upper class limits.

More-than type

Counts observations at or above successive lower class limits.

The chapter demonstrates both forms for constructing cumulative frequency distributions.

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9. Median of Grouped Data

The median represents the point that divides the distribution into two roughly equal parts.

For grouped data, first calculate:$$\frac{n}{2}$$

Then find the class whose cumulative frequency is just greater than n/2n/2.

This is the:$$\boxed{\text{Median class}}$$

The chapter uses this procedure explicitly.

Formula

$$\boxed{ \text{Median} = l+ \left( \frac{\frac n2-cf}{f} \right)h }$$

where:

• $l$ = lower limit of median class
• $n$ = total frequency
• $cf$ = cumulative frequency before the median class
• $f$ = frequency of median class
• $h$ = class width

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10. Steps for Finding Median

1. Find total frequency $n$.

2. Calculate $n/2$.

3. Construct cumulative frequencies.

4. Find the first cumulative frequency greater than $n/2$.

5. Its class is the median class.

6. Identify:$$l,\quad cf,\quad f,\quad h$$

7. Apply:$$\text{Median}=l+ \left(\frac{n/2-cf}{f}\right)h$$

8. Interpret the answer as the approximate middle value of the distribution.

For the chapter’s example with 53 observations, $n/2=26.5$, and the class whose cumulative frequency first exceeds this value is $60-70$, making it the median class.

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11. Less-Than Cumulative Frequency → Frequency

Sometimes a question gives cumulative frequencies instead of ordinary frequencies.

To recover a class frequency:$$\boxed{ f_{\text{current}} = cf_{\text{current}}-cf_{\text{previous}} }$$

For example, if the cumulative frequencies below two successive limits are 29 and 40:$$f=40-29=11$$

This is exactly the procedure used when converting the chapter’s height data into a frequency table.

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12. Continuous Class Intervals

For the median and mode formulas, the chapter emphasizes that class intervals should be continuous.

This becomes particularly important when data is given in intervals such as:$$118-126,\quad127-135,\quad136-144,\ldots$$

For measurements recorded to the nearest unit, boundaries can be adjusted to form continuous classes, e.g.:$$117.5-126.5,\quad126.5-135.5,\ldots$$

The chapter specifically gives this conversion as a requirement/hint for its median problem.

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13. Ogive — Cumulative Frequency Curve

An ogive is a graph representing cumulative frequency.

The chapter introduces ogives as cumulative-frequency curves.

For construction:

• use cumulative frequencies;
• use appropriate class boundaries/limits;
• ensure class intervals are continuous;
• the scales on the two axes do not have to be identical.

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14. Which Measure Should You Use?

Use Mean when:

• every observation should contribute;
• you want an overall average;
• you want to compare distributions.

Limitation: extreme values can significantly influence it.

Use Median when:

• you want a typical middle value;
• extreme observations are present;
• the individual extreme values should not dominate the result.

Use Mode when:

• you need the most frequently occurring value;
• you are interested in the most popular/common choice.

These distinctions are emphasized in the chapter’s discussion of practical applications.

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15. Empirical Relationship

The chapter gives the following relationship between the three measures:$$\boxed{3(\text{Median})=\text{Mode}+2(\text{Mean})}$$

It can also be rearranged depending on which quantity is required.

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16. Formula Sheet — Learn These

Class mark

$$\boxed{x_i=\frac{\text{lower limit}+\text{upper limit}}2}$$

Mean — Direct

$$\boxed{\bar{x}=\frac{\sum f_ix_i}{\sum f_i}}$$

Assumed Mean

$$\boxed{\bar{x}=a+\frac{\sum f_id_i}{\sum f_i}}$$

where$$\boxed{d_i=x_i-a}$$

Step Deviation

$$\boxed{\bar{x}=a+h\frac{\sum f_iu_i}{\sum f_i}}$$

where$$\boxed{u_i=\frac{x_i-a}{h}}$$

Mode

$$\boxed{ \text{Mode} = l+ \frac{f_1-f_0}{2f_1-f_0-f_2}h }$$

Median

$$\boxed{ \text{Median} = l+ \left(\frac{n/2-cf}{f}\right)h }$$

Empirical relation

$$\boxed{3M_d=M_o+2\bar{x}}$$

where $M_d$ is median and $M_o$ is mode.

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17. Most Important Exam Traps

Don’t confuse:

• Class mark → midpoint of a class.
• Modal class → class having maximum frequency.
• Median class → class whose cumulative frequency first exceeds $n/2$.
• Frequency → number of observations in one class.
• Cumulative frequency → running total of frequencies.

For median/mode:

• Check whether intervals are continuous.
• Use the correct lower limit.
• For median, use the cumulative frequency before the median class.
• For mode, use the frequencies immediately before and after the modal class.
• Keep $h$ as the class width.

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18. One-Minute Revision

Mean = average
Median = middle
Mode = most frequent

For grouped data:$$\boxed{\text{Mean}\rightarrow \sum fx}$$$$\boxed{\text{Median}\rightarrow n/2\rightarrow\text{median class}}$$$$\boxed{\text{Mode}\rightarrow\text{maximum frequency}\rightarrow\text{modal class}}$$