Statistics-Class-10-Find the median life time of a lamp-Exercise-13.3-Q-5-AP/TS/NCERT Syllubus

Interactive Central Tendency Calculator

Finding the Mean, Median, and Mode of Grouped Data

👇 The Problem

The following table gives the distribution of the life time of 400 neon lamps. We will find the mean, median, and mode life time of a lamp.

Median Calculation

🧠 Key Concepts for Median

• What is 'cumulative frequency'?
• How do you identify the 'median class' in grouped data?
• What is the formula for calculating the median of grouped data?
  \( \text{Median} = l + \left( \frac{\frac{N}{2} - cf}{f} \right) \times h \)

How can we find the Median?

Let's follow a clear plan. Click the button below to see the first step in our calculation.

Step 1: Create a Cumulative Frequency (cf) Table

Life time (in hours)	Number of lamps (f)	Cumulative Frequency (cf)
1500 - 2000	14	14
2000 - 2500	56	70
2500 - 3000	60	130
3000 - 3500	86	216
3500 - 4000	74	290
4000 - 4500	62	352
4500 - 5000	48	400
Total	N = 400

Step 2: Find the Median Class

Total lamps \(N = 400\). We calculate: \( \frac{N}{2} = \frac{400}{2} = 200 \). The class with a cumulative frequency just greater than 200 is the median class.

Therefore, the median class is 3000 - 3500.

Step 3: Identify variables for the formula

• \( l \) (lower limit of median class) = 3000
• \( h \) (class size) = 500
• \( f \) (frequency of median class) = 86
• \( cf \) (cumulative frequency of class before median class) = 130

Step 4: Substitute values into the Median Formula

\( \text{Median} = 3000 + \left( \frac{200 - 130}{86} \right) \times 500 = 3000 + \left( \frac{70}{86} \right) \times 500 \)

\( \text{Median} = 3000 + \frac{35000}{86} \approx 3000 + 406.98 \)

The median life time of a lamp is 3406.98 hours.

Mean Calculation (Step-Deviation Method)

🧠 Key Concepts for Mean

We'll use the Step-Deviation method, which is great for large numbers.

• 'Class Mark' (\(x_i\)) is the midpoint of a class interval.
• 'Assumed Mean' (\(A\)) is a guess, usually a central class mark.
• The formula is: \( \text{Mean} = A + \left( \frac{\sum f_i u_i}{\sum f_i} \right) \times h \), where \( u_i = \frac{x_i - A}{h} \).

How can we find the Mean?

Step 1: Create the Calculation Table

We need columns for Class Mark (\(x_i\)), \(u_i\), and \(f_i u_i\). Let's assume the mean \(A = 3250\).

Life time	\(f_i\)	\(x_i\)	\(u_i = \frac{x_i-3250}{500}\)	\(f_i u_i\)
1500-2000	14	1750	-3	-42
2000-2500	56	2250	-2	-112
2500-3000	60	2750	-1	-60
3000-3500	86	3250 (A)	0	0
3500-4000	74	3750	1	74
4000-4500	62	4250	2	124
4500-5000	48	4750	3	144
Total	\(\sum f_i = 400\)			\(\sum f_i u_i = 128\)

Step 2: Substitute values into the Mean Formula

\( \text{Mean} = 3250 + \left( \frac{128}{400} \right) \times 500 \)

\( \text{Mean} = 3250 + (0.32) \times 500 = 3250 + 160 \)

The mean life time of a lamp is 3410 hours.

Mode Calculation

🧠 Key Concepts for Mode

• The 'modal class' is the class with the highest frequency.
• The formula is: \( \text{Mode} = l + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h \)

How can we find the Mode?

Step 1: Find the Modal Class

Looking at the frequency column, the highest frequency is 86.

Life time (in hours)	Number of lamps (f)
2500 - 3000	60 (\(f_0\))
3000 - 3500	86 (\(f_1\))
3500 - 4000	74 (\(f_2\))

Therefore, the modal class is 3000 - 3500.

Step 2: Identify variables for the formula

• \( l \) (lower limit of modal class) = 3000
• \( h \) (class size) = 500
• \( f_1 \) (frequency of modal class) = 86
• \( f_0 \) (frequency of preceding class) = 60
• \( f_2 \) (frequency of succeeding class) = 74

Step 3: Substitute values into the Mode Formula

\( \text{Mode} = 3000 + \left( \frac{86 - 60}{2(86) - 60 - 74} \right) \times 500 \)

\( \text{Mode} = 3000 + \left( \frac{26}{172 - 134} \right) \times 500 = 3000 + \left( \frac{26}{38} \right) \times 500 \)

\( \text{Mode} = 3000 + \frac{13000}{38} \approx 3000 + 342.11 \)

The modal life time of a lamp is 3342.11 hours.

📊 Summary & Conclusion

Here are the three measures of central tendency for the lamp life times:

• Mean: 3410 hours (The average life time)
• Median: 3406.98 hours (The middle value; 50% of lamps last longer, 50% last shorter)
• Mode: 3342.11 hours (The most frequent life time interval)

The values are all quite close to each other, which suggests a fairly symmetrical data distribution without extreme outliers.