Chapter 6: Measures of Central Tendency – Questions & Answers

📌 Part A – 1 Mark Questions (Very Short Answer)

1. What is a measure of central tendency?
   A measure of central tendency is a single value that represents the entire data set.
2. Why do we use measures of central tendency? (Give one reason)
   To find a representative value for the whole data.
3. What is arithmetic mean?
   Arithmetic mean is the sum of all values divided by the number of values.
4. What is median?
   Median is the middle value when data is arranged in order.
5. What is mode?
   Mode is the value that appears most often in a data set.
6. Median is also called ________.
   Positional average.
7. Write the formula for arithmetic mean in individual series.
   Mean = ΣX / N, where ΣX = sum of values, N = number of items.
8. What are quartiles?
   Quartiles are values that divide a series into four equal parts.
9. What are deciles?
   Deciles are values that divide a series into ten equal parts.
10. What are percentiles?
    Percentiles are values that divide a series into one hundred equal parts.
11. Which quartile is equal to the median?
    The second quartile (Q2).
12. Which decile is equal to the median?
    The fifth decile (D5).
13. Which percentile is equal to the median?
    The fiftieth percentile (P50).
14. What is the formula to find mode when it is ill-defined?
    Mode = 3 Median – 2 Mean.
15. The sum of deviations of items from the arithmetic mean is always ________.
    Zero.
16. Which average is affected most by extreme values?
    Arithmetic mean.
17. Which average is best for qualitative data like intelligence or honesty?
    Median.
18. If mean = 25 and median = 20, find the approximate mode.
    Mode = 3(20) – 2(25) = 60 – 50 = 10.
19. Write the formula for combined mean.
    X̄12 = (N1 × X̄1 + N2 × X̄2) / (N1 + N2).
20. What is the formula for weighted arithmetic mean?
    Weighted Mean = ΣWX / ΣW, where W = weights, X = values.

📌 Part B – 2/4 Marks Questions (Short Answer)

1. Write any four qualities of a good average.

   • It should be rigidly defined.
   • It should be based on all observations.
   • It should be easy to calculate and understand.
   • It should not be affected by extreme values.
   • It should be capable of further mathematical treatment.
   • It should be representative of the entire data.

   (Any four points are sufficient)

2. List the importance of measures of central tendency.

   • To find a single representative value for the whole data.
   • To condense or summarise large data into one number.
   • To make comparisons between different data sets.
   • To help in further statistical analysis.

3. Differentiate between mean, median, and mode.

   • Mean: The arithmetic average. It uses all values. Affected by extreme values.
   • Median: The middle value. It divides data into two halves. Not affected by extremes.
   • Mode: The most frequent value. Useful for finding the most common item.

4. Calculate arithmetic mean for the data: 250, 275, 280, 300, 500, 250, 290, 350, 400, 375.

   Sum = 250 + 275 + 280 + 300 + 500 + 250 + 290 + 350 + 400 + 375 = 3270
   N = 10
   Mean = 3270 / 10 = 327

5. Find the median from the data: 15, 20, 25, 28, 16, 18, 17, 9, 11.

   Arrange in ascending order: 9, 11, 15, 16, 17, 18, 20, 25, 28
   N = 9 (odd)
   Median = size of (N+1)/2 th item = (9+1)/2 = 5th item = 17

6. Find the mode from the data: 41, 42, 45, 44, 45, 48, 50, 45, 47, 50, 56.

   Count the frequency of each number. 45 appears 3 times. No other number appears 3 times.
   So, Mode = 45

7. Explain the difference between quartiles, deciles, and percentiles.

   • Quartiles (Q1, Q2, Q3): Divide the data into 4 equal parts. Q2 is the median.
   • Deciles (D1 to D9): Divide the data into 10 equal parts. D5 is the median.
   • Percentiles (P1 to P99): Divide the data into 100 equal parts. P50 is the median.

   They are all positional values that divide a series into equal parts.

8. The average marks of 50 students was 44. Later it was found that a score of 36 was wrongly taken as 56. Find the correct mean.

   Incorrect total = 44 × 50 = 2200
   Correct total = 2200 – 56 + 36 = 2180
   Correct mean = 2180 / 50 = 43.6

9. The mean age of 40 students is 16 years and the mean age of another 60 students is 20 years. Find the combined mean age.

   Combined mean = (N1 × X̄1 + N2 × X̄2) / (N1 + N2)
   = (40 × 16 + 60 × 20) / (40 + 60)
   = (640 + 1200) / 100 = 1840 / 100 = 18.4 years

📌 Part C – 6/8 Marks Questions (Long Answer)

1. Define measures of central tendency. Explain the three main types of averages with formulas and examples.

   Measures of central tendency are single values that represent the center or typical value of a data set. They summarise the entire data into one number.

   The three main types are:

   1. Arithmetic Mean:

   • Formula (Individual series): Mean = ΣX / N, where X = values, N = number of items.
   • Example: For marks 10, 20, 30, 40, 50. Mean = (10+20+30+40+50)/5 = 150/5 = 30.
   • Formula (Discrete series): Mean = ΣfX / N, where f = frequency.
   • Formula (Continuous series): Mean = Σfm / N, where m = mid-value of class.

   2. Median:

   • Meaning: The middle value when data is arranged in order.
   • Formula (Individual series, odd N): Median = size of (N+1)/2 th item.
   • Formula (Continuous series): Median = L + [(N/2 – cf) / f] × i, where L = lower limit of median class, cf = cumulative frequency before median class, f = frequency of median class, i = class interval.
   • Example: For data 5, 7, 9, 11, 13. N=5. Median = (5+1)/2 = 3rd item = 9.

   3. Mode:

   • Meaning: The value that occurs most frequently.
   • Formula (Individual series): Identify the value with highest frequency.
   • Formula (Continuous series): Mode = L + [D1 / (D1 + D2)] × i, where D1 = frequency of modal class – frequency of pre-modal class, D2 = frequency of modal class – frequency of post-modal class.
   • Example: For data 2, 3, 3, 4, 5, 3. 3 appears three times, so Mode = 3.

2. Calculate the arithmetic mean from the following data showing marks of 70 students:

   Marks	No. of Students (f)
   0-10	5
   10-20	12
   20-30	15
   30-40	25
   40-50	8
   50-60	3
   60-70	2

   Solution:

   Marks	f	Mid-value (m)	fm
   0-10	5	5	25
   10-20	12	15	180
   20-30	15	25	375
   30-40	25	35	875
   40-50	8	45	360
   50-60	3	55	165
   60-70	2	65	130
   N = 70			Σfm = 2110

   Mean = Σfm / N = 2110 / 70 = 30.14 marks

3. Calculate the median for the following data:

   Daily Wages (₹)	No. of Workers
   20-25	14
   25-30	28
   30-35	33
   35-40	30
   40-45	20
   45-50	15
   50-55	13
   55-60	7

   Step 1: Calculate cumulative frequency (cf)

   Class	f	cf
   20-25	14	14
   25-30	28	42
   30-35	33	75
   35-40	30	105
   40-45	20	125
   45-50	15	140
   50-55	13	153
   55-60	7	160

   Step 2: Find N/2 N = 160, so N/2 = 80th item.

   Step 3: Identify median class. The class where cumulative frequency just exceeds 80 is 35-40 (cf = 105). So, median class = 35-40.

   Step 4: Apply formula.
   L = 35, cf = 75, f = 30, i = 5
   Median = L + [(N/2 – cf) / f] × i
   = 35 + [(80 – 75) / 30] × 5
   = 35 + (5/30) × 5
   = 35 + (25/30)
   = 35 + 0.83 = 35.83

   ∴ Median daily wage = ₹35.83

4. Explain the method of locating mode graphically. Also calculate mode from the given data:

   Graphical location of Mode:

   • Step 1: Draw a histogram of the given data.
   • Step 2: Identify the modal class (the bar with the highest height).
   • Step 3: Draw two diagonal lines inside the modal class bar. One line from the top-left corner of the modal bar to the top-right corner of the bar preceding it. Another line from the top-right corner of the modal bar to the top-left corner of the bar following it.
   • Step 4: From the point where these two diagonals intersect, draw a perpendicular line down to the X-axis. The point where this line meets the X-axis is the mode.

   Example: Calculate mode from the data:

   Income (₹)	No. of Families
   70-80	150
   80-90	140
   90-100	115
   100-110	95
   110-120	70
   120-130	60
   130-140	40

   Solution:
   Highest frequency is 150. Therefore, modal class = 70-80.
   L = 70, f1 = 150, f0 = 0 (no class before), f2 = 140
   D1 = f1 – f0 = 150 – 0 = 150
   D2 = f1 – f2 = 150 – 140 = 10
   i = 10
   Mode = L + [D1 / (D1 + D2)] × i
   = 70 + [150 / (150 + 10)] × 10
   = 70 + (150/160) × 10
   = 70 + (0.9375 × 10)
   = 70 + 9.375 = 79.375

5. Calculate Q1, Q3, D3, and P35 from the marks of 19 students: 18, 20, 25, 17, 9, 11, 23, 37, 38, 42, 36, 35, 8, 6, 11, 21, 20, 41, 35.

   Step 1: Arrange in ascending order.
   6, 8, 9, 11, 11, 17, 18, 20, 20, 21, 23, 25, 35, 35, 36, 37, 38, 41, 42.
   N = 19.

   Step 2: Calculate Q1 (First Quartile).
   Q1 = size of (N+1)/4 th item = (19+1)/4 = 20/4 = 5th item.
   5th item in the ordered list is 11.
   ∴ Q1 = 11 marks.

   Step 3: Calculate Q3 (Third Quartile).
   Q3 = size of 3(N+1)/4 th item = 3×(20)/4 = 60/4 = 15th item.
   15th item in the ordered list is 36.
   ∴ Q3 = 36 marks.

   Step 4: Calculate D3 (Third Decile).
   D3 = size of 3(N+1)/10 th item = 3×20/10 = 60/10 = 6th item.
   6th item in the ordered list is 17.
   ∴ D3 = 17 marks.

   Step 5: Calculate P35 (35th Percentile).
   P35 = size of 35(N+1)/100 th item = 35×20/100 = 700/100 = 7th item.
   7th item in the ordered list is 18.
   ∴ P35 = 18 marks.