Looking for the Statistics Class 10 formulas you need for a quick revision? This sheet covers the three measures of central tendency that the chapter extends to grouped data — the mean, the mode and the median — plus the class mark, cumulative frequency and the empirical relationship between the three measures.
All formulas come from the Rationalised NCERT Class 10 Mathematics textbook, Chapter 13. Each is grouped by topic, with a symbol table, when-to-use guidance and worked examples on fresh numbers. For the derivations and full explanations, see the Statistics Class 10 Notes; this page sits in the Class 10 Maths Formulas collection.
To verify any formula against the official text, open the chapter PDF (jemh113.pdf) at ncert.nic.in.
Statistics Class 10 Formulas at a Glance
Every formula on this page appears in the table below, matched to what it finds. Symbol meanings and validity conditions follow in the next sections.
| Purpose | Formula |
|---|---|
| Mean of grouped data — direct method | \( \bar{x} = \frac{\sum f_i x_i}{\sum f_i} \) |
| Class mark of a class interval | \( \frac{\text{Upper class limit} + \text{Lower class limit}}{2} \) |
| Mean of grouped data — assumed mean method | \( \bar{x} = a + \frac{\sum f_i d_i}{\sum f_i} \), where \( d_i = x_i – a \) |
| Mean of grouped data — step-deviation method | \( \bar{x} = a + h\left(\frac{\sum f_i u_i}{\sum f_i}\right) \), where \( u_i = \frac{x_i – a}{h} \) |
| Mode of grouped data (most frequent value) | \( \text{Mode} = l + \left(\frac{f_1 – f_0}{2f_1 – f_0 – f_2}\right) \times h \) |
| Median of grouped data (middle observation) | \( \text{Median} = l + \left(\frac{\frac{n}{2} – cf}{f}\right) \times h \) |
| Median of ungrouped data (recalled from Class IX) | \( \left(\frac{n+1}{2}\right)\text{th} \) observation when \( n \) is odd; average of the \( \frac{n}{2}\text{th} \) and \( \left(\frac{n}{2}+1\right)\text{th} \) observations when \( n \) is even |
| Empirical relationship between the three measures | \( 3 \times \text{Median} = \text{Mode} + 2 \times \text{Mean} \) |
All Formulas, Grouped by Topic
Formulas are grouped under the sub-topics the NCERT chapter itself uses: mean, mode and median of grouped data, plus the empirical relation between them.
Mean of Grouped Data
Each class is replaced by its class mark — the mid-point that stands for the whole class, since the frequency of a class is assumed to be centred there (NCERT, p. 173).
\[ \text{Class mark} = \frac{\text{Upper class limit} + \text{Lower class limit}}{2} \]
Direct method (NCERT, p. 172):
\[ \bar{x} = \frac{\sum f_i x_i}{\sum f_i} \]
Assumed mean method — subtract a chosen value \( a \) from each \( x_i \) before multiplying (NCERT, p. 176):
\[ \bar{x} = a + \frac{\sum f_i d_i}{\sum f_i}, \qquad d_i = x_i – a \]
Step-deviation method — divide each deviation by the class size \( h \) to shrink the numbers further (NCERT, p. 177):
\[ \bar{x} = a + h\left(\frac{\sum f_i u_i}{\sum f_i}\right), \qquad u_i = \frac{x_i – a}{h} \]
All three methods return the same mean — they differ only in the amount of arithmetic they save.
Mode of Grouped Data
The modal class is the class with the maximum frequency. The mode is a value inside the modal class, obtained from (NCERT, p. 184):
\[ \text{Mode} = l + \left(\frac{f_1 – f_0}{2f_1 – f_0 – f_2}\right) \times h \]
Here \( h \) is the class size, assuming all class sizes are equal. The class intervals must be made continuous before the formula is applied, and the chapter restricts problems to a single mode.
Median of Grouped Data
For ungrouped data, the median is the middle observation after arranging the values in ascending order (NCERT, p. 189):
- If \( n \) is odd, the median is the \( \left(\frac{n+1}{2}\right)\text{th} \) observation.
- If \( n \) is even, the median is the average of the \( \frac{n}{2}\text{th} \) and \( \left(\frac{n}{2}+1\right)\text{th} \) observations.
For grouped data, first find \( \frac{n}{2} \) and locate the median class — the class whose cumulative frequency is greater than and nearest to \( \frac{n}{2} \). Then (NCERT, p. 194):
\[ \text{Median} = l + \left(\frac{\frac{n}{2} – cf}{f}\right) \times h \]
The cumulative frequency of a class is the sum of the frequencies of all classes up to and including it. Built using upper limits it gives a less than type distribution; using lower limits, a more than type distribution (NCERT, pp. 192-193). When data is supplied in the less-than form, recover each class frequency by subtracting successive cumulative frequencies (NCERT, p. 195).
Relation Between Mean, Median and Mode
\[ 3 \times \text{Median} = \text{Mode} + 2 \times \text{Mean} \]
The chapter gives this empirical relationship between the three measures of central tendency (NCERT, p. 198). Use it to check that your three computed values are consistent with one another.
What Each Symbol Means
| Symbol | What it means | Unit / nature |
|---|---|---|
| \( \bar{x} \) | Mean of the grouped data | Same unit as the observations (marks, rupees, cm, …) |
| \( x_i \) | Class mark (mid-point) of the \( i \)-th class | Same unit as the observations |
| \( f_i \) | Frequency of the \( i \)-th class | A count (no unit) |
| \( n \) | Total number of observations, \( n = \sum f_i \) | A count (no unit) |
| \( a \) | Assumed mean — a chosen \( x_i \), usually near the centre | Same unit as the observations |
| \( d_i \) | Deviation of \( x_i \) from \( a \): \( d_i = x_i – a \) (can be negative) | Same unit as the observations |
| \( u_i \) | Step-deviation: \( u_i = \frac{x_i – a}{h} \) | Dimensionless (a ratio) |
| \( h \) | Class size (width of a class interval) | Same unit as the observations |
| \( l \) | Lower limit of the modal class or the median class | Same unit as the observations |
| \( f_1 \) | Frequency of the modal class | A count (no unit) |
| \( f_0 \) | Frequency of the class preceding the modal class | A count (no unit) |
| \( f_2 \) | Frequency of the class succeeding the modal class | A count (no unit) |
| \( f \) | Frequency of the median class | A count (no unit) |
| \( cf \) | Cumulative frequency of the class preceding the median class | A count (no unit) |
When to Use Each Formula
The chapter’s remark on choosing a method (NCERT, p. 180) is the practical guide:
| Formula | Reach for it when… |
|---|---|
| Class mark | Converting a class interval into one representative value before any mean calculation |
| Direct method | \( x_i \) and \( f_i \) are small, so the products \( f_i x_i \) are easy to add |
| Assumed mean method | \( x_i \) and \( f_i \) are numerically large; choose \( a \) as a central \( x_i \) |
| Step-deviation method | The deviations \( d_i \) share a common factor; divide by \( h \). Works with unequal class sizes if \( h \) is a suitable divisor of all the \( d_i \)’s |
| Mode formula | You need the most frequent value — locate the modal class first |
| Median formula | You need the middle observation — locate the median class first via cumulative frequency |
| Empirical relationship | Cross-checking that the mean, median and mode computed from the same table are consistent |
When deciding which measure you actually need, the chapter suggests (NCERT, p. 197):
- Mean — uses every observation and allows comparing distributions, but extreme values pull it away from most of the data.
- Median — better for a typical value when extreme observations are present.
- Mode — best when you want the most frequent or most popular value.
Worked Examples
The three examples below apply the main formulas on fresh numbers. For the textbook’s own questions, work through the Statistics NCERT Solutions.
Worked Example 1: Mean by the Direct Method
The marks (out of 50) of 30 students are grouped below. Find the mean marks.
| Marks | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 |
|---|---|---|---|---|---|
| Number of students | 3 | 7 | 10 | 7 | 3 |
Step 1: The class marks are \( x_i = 5, 15, 25, 35, 45 \).
The frequencies are small, so the direct method is the quickest choice.
Step 2: Sum the products \( f_i x_i \):
\[ \sum f_i x_i = 3(5) + 7(15) + 10(25) + 7(35) + 3(45) = 15 + 105 + 250 + 245 + 135 = 750 \]
Step 3: Apply the direct-method formula with \( \sum f_i = 30 \):
\[ \bar{x} = \frac{\sum f_i x_i}{\sum f_i} = \frac{750}{30} = 25 \]
Final answer: The mean marks are 25 out of 50. The answer lies between the smallest class mark (5) and the largest (45), as a mean must.
Worked Example 2: Mode of Grouped Data
The time 50 students spent on homework in an evening is grouped below. Find the mode.
| Time (minutes) | 0-20 | 20-40 | 40-60 | 60-80 | 80-100 |
|---|---|---|---|---|---|
| Number of students | 5 | 12 | 18 | 9 | 6 |
- Step 1: The maximum frequency is 18, so the modal class is 40-60.
- Step 2: Read the values: \( l = 40 \), \( h = 20 \), \( f_1 = 18 \), \( f_0 = 12 \), \( f_2 = 9 \).
- Step 3: Substitute into the mode formula:
\[ \text{Mode} = 40 + \left(\frac{18 – 12}{2(18) – 12 – 9}\right) \times 20 = 40 + \frac{6}{15} \times 20 = 40 + 8 = 48 \]
Final answer: The modal time spent on homework is 48 minutes — inside the modal class 40-60, as it must be.
Worked Example 3: Median of Grouped Data
The heights (in cm) of 40 plants are grouped below. Find the median height.
| Height (cm) | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 |
|---|---|---|---|---|---|
| Number of plants | 5 | 8 | 14 | 9 | 4 |
Step 1: Add a cumulative frequency column: 5, 13, 27, 36, 40.
Here \( n = 40 \), so \( \frac{n}{2} = 20 \).
Step 2: The first cumulative frequency greater than 20 is 27, so the median class is 30-40.
Read \( l = 30 \), \( cf = 13 \), \( f = 14 \), \( h = 10 \).
Step 3: Apply the median formula:
\[ \text{Median} = 30 + \left(\frac{20 – 13}{14}\right) \times 10 = 30 + 5 = 35 \]
Final answer: The median height is 35 cm — about half the plants are shorter and half are taller.
Common Mistakes to Avoid
The table lists the errors students most often make when applying these formulas, with the correction and a quick self-check.
| Mistake | Correct rule | How to check your answer |
|---|---|---|
| Using a class endpoint as the class mark | Class mark \( = \frac{\text{Upper limit} + \text{Lower limit}}{2} \) | The class mark must lie exactly midway between the two limits. |
| Applying the mode or median formula to discontinuous classes | Make the classes continuous first — e.g. 117.5-126.5, 126.5-135.5, … | Check that the upper limit of each class equals the lower limit of the next. |
| Swapping \( f_0 \) and \( f_2 \) in the mode formula | \( f_0 \) is the frequency of the class before the modal class; \( f_2 \) is the class after it. | Check that \( f_1 \) is the largest frequency and that \( f_0 \), \( f_2 \) are its immediate neighbours in the table. |
| Using the median class’s own cumulative frequency as \( cf \) | \( cf \) is the cumulative frequency of the class preceding the median class. | Check \( cf \lt \frac{n}{2} \lt cf + f \). |
| Treating less-than entries as ordinary frequencies | Subtract successive cumulative frequencies: e.g. \( 11 – 4 = 7 \) gives the frequency of the class 140-145. | Check that the recovered frequencies add up to \( n \). |
Frequently Asked Questions
Which mean method should I choose — direct, assumed mean or step-deviation?
Use the direct method when the values \( x_i \) and the frequencies \( f_i \) are small. For numerically large values, use the assumed mean method or the step-deviation method; pick step-deviation when the deviations share a common factor. All three give exactly the same mean (NCERT, p. 180). For a chapter-by-chapter index, browse the Maths Formulas collection.
What is the difference between the modal class and the mode?
The modal class is the class interval with the maximum frequency — you can spot it straight from the table. The mode is a single value inside that class, produced by the formula, and it is usually not equal to the class mark of the modal class.
Why must class intervals be continuous before finding the mode or median?
The formulas assume the frequency is spread continuously across each class. The chapter’s note to the reader states that class intervals must be continuous before applying the mode and median formulas, and that the same condition applies when drawing an ogive.
How do I find the median when the data is given in the less-than form?
Convert the cumulative table into a frequency table by subtracting successive entries — for example, 11 girls are shorter than 145 cm and 4 are shorter than 140 cm, so the class 140-145 has frequency \( 11 – 4 = 7 \). Then proceed with the usual median formula (NCERT, p. 195).
Reference: NCERT Class 10 Mathematics textbook, chapter Statistics.
Explore Class 10 Maths Formulas
- Previous: Surface Areas and Volumes
- Next: Probability
More for this chapter:
Related chapters:
- Real Numbers notes
- Polynomials notes
- Pair of Linear Equations in Two Variables notes