Mean, median, mode — calculation for raw and grouped data

Overview

The three measures of central tendency tell us where the "center" of a dataset lives. Each answers a different question: mean is the arithmetic average, median is the middle value when sorted, and mode is the most frequent value. Understanding when to use each and how they differ for raw vs. grouped data is crucial for interpreting real-world datasets.

Core Intuition

Raw Data (Ungrouped)

Mean (Arithmetic Average)

Median (Middle Value)

Mode (Most Frequent)

Grouped Data (Frequency Distribution)

When data is organized into class intervals (e.g., 0-10, 10-20), we can't see individual values. We use class marks (midpoints) and frequencies (counts).

Figure — Mean, median, mode — calculation for raw and grouped data

Mean for Grouped Data

Worked example Example 4: Student test scores (grouped)

Class Interval	Frequency ( $f$ )	Class Mark ( $x$ )	$f \cdot x$
0-20	5	10	50
20-40	8	30	240
40-60	12	50	600
60-80	10	70	700
80-100	5	90	450
Total	$N=40$	—	2040

Step 1: Calculate class marks

For0-20: $x = \frac{0+20}{2} = 10$ Why midpoint? We're assuming uniform distribution within the class.

Step 2: Multiply $f \times x$ for each class

For 0-20: $5 \times 10 = 50$ Why? 5 students assumed to be at 10 contribute $5 \times 10$ .

Step 3: Sum all $f \cdot x = 2040$ Step 4: Mean = $\frac{2040}{40} = 51$ marks Interpretation: The average score is 51.

Median for Grouped Data

Formula Median for Grouped Data

$\text{Median} = L + \left( \frac{\frac{N}{2} - CF}{f} \right) \times h$ where:

$L$ = lower boundary of the median class
$N$ = total frequency
$CF$ = cumulative frequency of the class before the median class
$f$ = frequency of the median class
$h$ = class width (upper limit − lower limit)

DERIVATION: The median is at position $\frac{N}{2}$ . We find which class contains this position (median class). Then we interpolate linearly within that class:

We need to "travel" $\left(\frac{N}{2} - CF\right)$ more observations into the median class
The class has $f$ observations spread over width $h$
So we go $\frac{\frac{N}{2} - CF}{f} \times h$ units from the lower boundary $L$

Worked example Example 5: Median from grouped data

Using the same table from Example 4, add cumulative frequency:

Class	Frequency ( $f$ )	Cumulative Frequency (CF)
0-20	5	5
20-40	8	13
40-60	12	25 ← median class
60-80	10	35
80-100	5	40

Step 1: Find $\frac{N}{2} = \frac{40}{2} = 20$ Why? The median is the 20th observation.

Step 2: Locate median class

CF up to 20-40 is 13(< 20)
CF up to 40-60 is 25 (≥ 20) ← median class is 40-60 Why? The 20th value falls in this class.

Step 3: Apply formula

$L = 40$ (lower boundary of 40-60)
$CF = 13$ (cumulative before median class)
$f = 12$ (frequency of median class)
$h = 60 - 40 = 20$

$\text{Median} = 40 + \left( \frac{20 - 13}{12} \right) \times 20 = 40 + \frac{7}{12} \times 20 = 40 + 11.67 = 51.67$

Why this step? We need 7 more observations (20 − 13) out of 12 in the class. That's $\frac{7}{12}$ of the class width (20), so we add $11.67$ to the lower boundary.

Median ≈ 51.67 marks

Mode for Grouped Data

Worked example Example 6: Mode from grouped data

From Example 4 table, modal class = 40-60 (highest frequency 12)

Step 1: Identify values

$L = 40$
$f_1 = 12$ (frequency of 40-60)
$f_0 = 8$ (frequency of 20-40, the class before)
$f_2 = 10$ (frequency of 60-80, the class after)
$h = 20$

Step 2: Apply formula $\text{Mode} = 40 + \left( \frac{12 - 8}{2(12) - 8 - 10} \right) \times 20$ $= 40 + \left( \frac{4}{24 - 18} \right) \times 20 = 40 + \frac{4}{6} \times 20 = 40 +13.33 = 53.33$

Why this step? The modal class 40-60 has midpoint 50. The left gap $f_1-f_0 = 12-8 = 4$ is larger than the right gap $f_1-f_2 = 12-10 = 2$ . Because the frequency drops off more sharply on the right (from 12 to 10) than the rise on the left, the peak leans toward the right boundary — which is why the mode (≈ 53.33) lies to the right of the midpoint 50.

Mode ≈ 53.33 marks

Common Mistakes

Step-by-Step Algorithms

Raw Data

Mean: Sum all values → divide by count
Median: Sort data → if odd, pick middle; if even, average two middle values
Mode: Count frequencies → pick most frequent (or identify none/multiple modes)

Grouped Data

Mean: Compute class marks → multiply by frequencies → sum → divide by $N$
Median: Compute cumulative frequency → find median class (where $CF \geq \frac{N}{2}$ ) → apply interpolation formula
Mode: Find modal class (highest $f$ ) → apply interpolation formula using neighbors

When to Use Each Measure

Measure	Best For	Limitation
Mean	Symmetric data, when you need total/sum	Sensitive to outliers
Median	Skewed data, outliers present	Ignores actual values, only position
Mode	Categorical data, finding "typical" value	May not exist or be multiple

Feynman Explanation

Recall Explain to a 12-year-old

Imagine you and4 friends got these chocolate bars:2, 3, 5, 6 and 9 bars.

Mean (average): If you pooled all chocolates (2+3+5+6+9 = 25) and shared equally, everyone gets 25÷5 = 5 bars. It's fair, but if one friend got 100 bars (lucky!), suddenly the average becomes 20, even though most of you still have fewer than 10. Mean is pulled by extremes.

Median (middle): Line up in order: {2, 3, 5, 6, 9}. The person in the middle has 5 bars. Even if the last friend had 1000 bars, the middle person still has 5. Median doesn't care about extremes!

Mode (most popular): If 3 friends had 5 bars,1 had 2, 1 had 9, then 5 is the mode—it appears most. It tells you what's "normal" in your group.

For grouped data (like "5 friends have 0-5 bars, 3 friends have 5-10 bars"), you can't see exact numbers. So you pretend everyone in the 0-5 group has 2.5 bars (the middle of0 and 5) and calculate. It's like estimating!

Mnemonic

Connections Measures of Dispersion — after finding the center, we measure spread (range, variance, standard deviation)

Cumulative Frequency Graphs — ogives help visualize median and quartiles
Frequency Distributions — foundation for grouped data calculations
Box Plots — visual summary using median and quartiles
Skewness — relationship between mean, median, mode in skewed distributions (mean > median > mode for right skew)
Weighted Mean — generalization where each value has a weight (similar to $f \cdot x$ in grouped data)

Flashcards

#flashcards/maths

What is the formula for mean of raw data? :: $\bar{x} = \frac{\sum x_i}{n}$ where $n$ is the count of observations.

How do you find the median for an odd number of raw data values?

Sort the data, then pick the

\left(\frac{n+1}{2}\right)^{\text{th}}

term.

How do you find the median for an even number of raw data values?

Sort the data, then average the

\left(\frac{n}{2}\right)^{\text{th}}

and

\left(\frac{n}{2}+1\right)^{\text{th}}

terms.

What is the mode of a dataset?

The value that appears most frequently. A dataset can have no mode, one mode, or multiple modes (e.g., bimodal if two values tie for highest frequency).

What is the class mark (midpoint) of a class interval 30-40?

\frac{30+40}{2} = 35

Formula for mean of grouped data?

\bar{x} = \frac{\sum f \cdot x}{N}

where

f

is frequency,

x

is class mark,

N

is total frequency.

Formula for median of grouped data?

\text{Median} = L + \left( \frac{\frac{N}{2} - CF}{f} \right) \times h

where

L

is lower boundary of median class,

CF

is cumulative frequency before it,

f

is its frequency,

h

is class width.

How do you identify the median class in grouped data?

Find the class where cumulative frequency first equals or exceeds

\frac{N}{2}

Formula for mode of grouped data?

\text{Mode} = L + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h

where

L

is lower boundary of modal class,

f_1

is its frequency,

f_0

is before,

f_2

is frequency after,

h

is class width.

How do you identify the modal class?

The class with the highest frequency.

Why must you sort data before finding the median?

Median is defined as the middle value in ordered data; without sorting, you cannot identify the true middle position.

Why do we use class marks instead of class boundaries for mean in grouped data?

We assume all values in a class are uniformly distributed, so the midpoint (class mark) is the best estimate of the typical value in that class.

If mean< median < mode, the distribution is __ skewed.

Left-skewed (negatively skewed).

If mean > median > mode, the distribution is ___ skewed.

Right-skewed (positively skewed).

Why is median better than mean for data with outliers?

Median depends only on the middle position(s), not actual values, so extreme outliers don't distort it.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho yaar, central tendency ka matlab hai dataset ka "center" kahan hai — aur yahan sabse important baat ye hai ki humein teen alag-alag averages ki zaroorat kyun padti hai. Mean total ko sabme barabar baant deta hai, lekin ye extreme values (outliers) se kheench jaata hai. Jaise agar ek student ne 95 marks laaye aur baaki 60-70 range mein hain, toh mean upar chala jaata hai. Median middle wala value hota hai jab data sort kiya jaaye — ye outliers ko ignore karta hai, isliye zyada "resistant" hai. Aur mode batata hai ki sabse frequent ya "typical" value kaun si hai. Har dataset ki apni shape hoti hai, isliye ek hi average pe bharosa karna galat ho sakta hai.

Calculation ka logic simple hai — mean ke liye saare values ka sum karke count se divide karo (matlab total ko equally distribute karna). Median ke liye pehle data ko ascending order mein sort karo, phir odd count hone par middle term le lo, aur even count hone par do middle terms ka average nikaalo — kyunki tab ek single middle value hoti hi nahi. Mode ke liye bas frequencies count karo aur jo sabse zyada baar aaye wo mode. Yaad rakhna, kabhi-kabhi do values tie ho jaati hain (bimodal), toh dono ko mode maanenge.

Ab grouped data mein thoda twist hai — jab data class intervals (jaise 0-10, 10-20) mein diya ho, tab humein individual values dikhti hi nahi. Isliye har class ka class mark (midpoint = lower+upper limit ka average) nikaalte hain aur usko us class ki frequency se multiply karte hain. Ye isliye important hai kyunki real-world mein zyadatar data grouped form mein hi milta hai — survey results, exam scores ki ranges, income brackets sab. Toh in dono cases (raw aur grouped) ko samajhna aapke liye practical data interpret karne mein bahut kaam aayega, aur exams mein bhi ye topic guaranteed aata hai.

Mean, median, mode — calculation for raw and grouped data

Overview

Core Intuition

Raw Data (Ungrouped)

Mean (Arithmetic Average)

Median (Middle Value)

Mode (Most Frequent)

Grouped Data (Frequency Distribution)

Mean for Grouped Data

Median for Grouped Data

Mode for Grouped Data

Common Mistakes

Step-by-Step Algorithms

Raw Data

Grouped Data

When to Use Each Measure

Feynman Explanation

Mnemonic

Connections Measures of Dispersion — after finding the center, we measure spread (range, variance, standard deviation)

Flashcards

Concept Map

Hinglish (regional understanding)

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Basic Data & Probability

Connections