CBSE Class 10 Mathematics Statistics MCQs Set 17

Answer: (b) 15

Explanation:
1. Sum of the four numbers = 4 × 9 = 36.
2. 5 + 7 + 9 + x = 36, so x = 15.

Teacher's Note:
1. Mean = sum ÷ number of observations.
2. Check: (5 + 7 + 9 + 15)/4 = 9.

Answer: (a) 45

Explanation:
1. Total marks = 4 × 50 = 200.
2. 36 + 44 + 75 + x = 200, so x = 45.

Teacher's Note:
1. Average × number of subjects = total.
2. Check: (36 + 44 + 75 + 45)/4 = 50.

Answer: (d) 27.5

Explanation:
1. Step deviation: x̄ = a + h × (Σfᵢuᵢ/Σfᵢ) with a = 20 and h = 10.
2. x̄ = 20 + 10 × 30/40 = 27.5.

Teacher's Note:
1. Read a and h from uᵢ = (xᵢ − a)/h.
2. The step-deviation method reduces the size of the numbers.

Answer: (b) 18

Explanation:
1. Adding a constant to every observation adds the same constant to the mean.
2. New mean = 13 + 5 = 18.

Teacher's Note:
1. Mean shifts by the added amount; the spread is unchanged.
2. Check: total increases by 100, and 100/20 = 5.

Answer: (c) 190

Explanation:
1. Arrange in ascending order: 20, 35, 50, 80, 190, 340, 520, 800, 1210.
2. There are 9 observations, so the median is the 5th value, 190.

Teacher's Note:
1. Always sort the data before picking the middle value.
2. For n odd the median is the ((n + 1)/2)th value.

Answer: (b) 13th

Explanation:
1. For n = 25 (odd), the median is the ((n + 1)/2)th = 13th observation.
2. The 13th has 12 observations on each side, whether the order is ascending or descending.

Teacher's Note:
1. The median is the middle value regardless of ascending or descending order.
2. For even n it is the mean of the (n/2)th and (n/2 + 1)th values.

Answer: (b) 21

Explanation:
1. There are 10 observations, so the median is the mean of the 5th and 6th terms: (x + 2 + x + 4)/2 = 24.
2. x + 3 = 24, so x = 21.

Teacher's Note:
1. With an even number of observations, average the two middle ones.
2. Check: 23 and 25 give 24.

Answer: (a) zero

Explanation:
1. The median of 10 observations is the mean of the 5th and 6th entries.
2. Increasing the 7th entry does not change the 5th and 6th (the order is preserved), so the median is unchanged.

Teacher's Note:
1. Only the middle entries affect the median.
2. Changing a value on one side of the middle without crossing it leaves the median unaltered.

Answer: (b) 4

Explanation:
1. Frequencies: 4 occurs three times, 3 occurs twice and every other value once.
2. The mode is the most frequent value, 4.

Teacher's Note:
1. Mode = value with the highest frequency.
2. A data set can have more than one mode.

Answer: (c) 9.2

Explanation:
1. Empirical relation: Mode = 3 Median − 2 Mean.
2. Mode = 3 × 9 − 2 × 8.9 = 27 − 17.8 = 9.2.

Teacher's Note:
1. Memorise the empirical relation between the three averages.
2. It holds for moderately skewed distributions.

Answer: (c) 100

Explanation:
1. Mode = 3 Median − 2 Mean, so 80 = 3 Median − 220.
2. 3 Median = 300 and Median = 100.

Teacher's Note:
1. Rearrange the empirical formula to isolate the median.
2. Median = (Mode + 2 Mean)/3.

Answer: (a) 3

Explanation:
1. 6 occurs 3 times and 7 occurs 3 times, so there are two modes unless 2k + 1 = 7.
2. For 7 to be the unique mode, 2k + 1 = 7, so k = 3.

Teacher's Note:
1. The mode is 7 only if 7 occurs more often than any other value.
2. Adding another 7 makes it appear 4 times.

Answer: (d) Mean = Median = Mode

Explanation:
1. In a symmetrical distribution the data are balanced about the centre.
2. So the mean, median and mode all coincide.

Teacher's Note:
1. Skewness separates the three measures.
2. A perfectly symmetric single-peaked distribution has all three equal.

Answer: (b) Median

Explanation:
1. At the intersection, the cumulative frequency below equals the cumulative frequency above.
2. That happens at the middle of the data, which is the median.

Teacher's Note:
1. The two ogives meet where half the observations lie on each side.
2. The y-coordinate of the point is N/2.

Answer: (d) 15.5

Explanation:
1. The x-coordinate of the intersection is the median.
2. So the median is 15.5.

Teacher's Note:
1. The y-coordinate 20 is a cumulative frequency, not the median.
2. Remember: median is read on the horizontal axis.

Answer: (c) 32

Explanation:
1. The smallest known value is 12 and the largest is 18.
2. If x were the maximum: x − 12 = 20, so x = 32. If x were the minimum: 18 − x = 20 gives x = −2, which is not allowed since x > 0.
3. So x = 32.

Teacher's Note:
1. Range = maximum − minimum.
2. Test both possibilities for where x lies.

Answer: (b) 12

Explanation:
1. If x lies between 14 and 35 the range is 35 − 14 = 21, not 23.
2. So x is either the new minimum (35 − x = 23, x = 12) or the new maximum (x − 14 = 23, x = 37).
3. The least possible value is 12.

Teacher's Note:
1. Check both ends of the data set.
2. Pick the smaller of the two valid values.

Answer: (b) 33.1

Explanation:
1. Total width of nine classes = 9 × 2.5 = 22.5.
2. Upper boundary of the highest class = 10.6 + 22.5 = 33.1.

Teacher's Note:
1. The classes are consecutive, so the widths add up.
2. Do not use only eight widths.

Answer: (b) 4

Explanation:
1. Sum = 0 + 0 + 2 + 2 + 3 + 3 + 3 + 4 + 5 + 5 + 5 + 5 + 6 + 6 + 7 + 8 = 64.
2. Mean = 64/16 = 4.

Teacher's Note:
1. Add the marks carefully in groups.
2. Mean = sum ÷ number of students.

Answer: (d) 90

Explanation:
1. Adding 45 to every observation adds 45 to the mean.
2. New mean = 45 + 45 = 90.

Teacher's Note:
1. Adding a constant shifts the mean by that constant.
2. The number of observations does not matter.

Answer: (d) 35

Explanation:
1. Total of 5 numbers = 135; total of the remaining 4 = 100.
2. Excluded number = 135 − 100 = 35.

Teacher's Note:
1. Compute both totals and subtract.
2. Check: (100 + 35)/5 = 27.

Answer: (b) 1

Explanation:
1. Total frequency = 15 and Σfx = 4 + 20 + 42 + 24 = 90, so the mean = 6.
2. The highest frequency is 6, at x = 7, so the mode = 7.
3. Mode − mean = 7 − 6 = 1.

Teacher's Note:
1. Mode is the value with the highest frequency.
2. Mean = Σfx/Σf.

Answer: (d) 20°C

Explanation:
1. Total for the week = 7 × 24 = 168.
2. Mon–Thu total = 88 and Thu–Sun total = 100; adding them counts Thursday twice: 188.
3. Thursday = 188 − 168 = 20°C.

Teacher's Note:
1. Thursday appears in both groups, so it is counted twice.
2. Subtract the weekly total from the sum of the two group totals.