Statistics (Mean, Median, Mode) Questions and Answers

Answer: C

The correct assertions are P and R.

• P is true: adding a constant to every entry adds it to the mean.
• Q is false: adding a constant shifts every entry and the mean equally, so the deviations from the mean, and the standard deviation, are unchanged.
• R is true: doubling every entry doubles the mean.
• S is false: doubling every entry doubles every deviation, so the standard deviation doubles.

Every other option includes Q or S, so P, R is the only fully correct choice.

Answer: B

The correct order is mode \(\gt\) median \(\gt\) mean.

There are \(3 + 9 + 11 + 7 + 14 + 2 + 4 = 50\) students.

• Mode: the highest frequency, 14, is at 7 marks.
• Median: the 25th and 26th values. Cumulative frequencies are 3, 12, 23, 30, ..., so both fall at 6 marks: median 6.
• Mean: \((9 + 36 + 55 + 42 + 98 + 16 + 36)/50 = 292/50 = 5.84\).

So 7 \(\gt\) 6 \(\gt\) 5.84. The other options put the mean above the median or the median above the mode, which these numbers contradict.

Answer: C

The weighted mean is about 15.7.

Multiply each value by its weight, add, and divide by the total weight:

\[ \begin{aligned} \textstyle\sum wx &= 20 + 60 + 180 + 72 + 140 = 472 \\ \textstyle\sum w &= 2 + 5 + 12 + 4 + 7 = 30 \end{aligned} \]

\[ \bar{x}_{w} = 472/30 = 15.73 \]

The simple mean of the five values is \(75/5 = 15\), which ignores the weights. 15.2, 15.6 and 14.3 come from arithmetic slips.

Answer: B

The median is 17.

Arrange the nine values in order:

5, 8, 9, 13, 17, 19, 26, 26, 29

With an odd number of values (9), the median is the middle one, the (9 + 1)/2 = 5th value, which is 17.

13 and 19 are the values either side of the middle, and 18 would be the median only if there were an even number of values with 17 and 19 in the middle, which there are not.

Answer: A

The median is 52.

Arrange the nine values in order:

33, 36, 45, 49, 52, 60, 61, 72, 72

With an odd number of values (9), the median is the middle one, the 5th, which is 52.

49 is the 4th value and 60 the 6th, so choosing either means miscounting the position. 61 is what you get by taking the 5th value of the unsorted list without arranging it first.

Answer: B

The median is 9.

Arrange the seven values in order: 5, 6, 8, 9, 12, 13, 17. With an odd number of values the median is the middle one, the 4th, which is 9.

8 and 12 are the values either side of the middle, and 17 is the largest value. Taking the middle of the list in its printed order (8) without sorting is the usual mistake.

Answer: D

The range is 6 and the coefficient of range is infinite, so the answer is 6 and \(\infty\).

With the largest value \(L = 3\) and the smallest \(S = -3\):

\[ \begin{aligned} \text{Range} &= L - S = 3 - (-3) = 6 \\ \text{Coefficient} &= \frac{L - S}{L + S} = \frac{6}{0} \end{aligned} \]

Division by zero has no finite value, so the coefficient is taken as infinite, which is why this measure is meaningless for data centred on zero. Options giving the range as 0 forget the sign of -3, and '6 and 0' wrongly treats \(\frac{6}{0}\) as 0.

Answer: A

About 68% of the data lie within one standard deviation of the mean.

By the empirical (68-95-99.7) rule for a normal distribution, about 68% of values lie within one standard deviation of the mean, 95% within two, and 99.7% within three.

So 95% and 99.7% are the two- and three-standard-deviation figures, and 50% is the share on either side of the mean.

The rule holds exactly only for normally distributed data; for other distributions Chebyshev's inequality gives weaker guarantees.

Answer: A

Zero skewness indicates that The dataset is symmetric.

Skewness measures asymmetry. A skewness of 0 means the distribution is balanced about its mean, with neither tail longer.

A normal distribution has zero skewness, but so do many other symmetric distributions, so zero skewness does not prove normality. Negative skewness means a longer left tail, and skewness says nothing directly about outliers.

Answer: A

The z-score tells you how far a data point is from the Mean in terms of Standard deviations.

\(z = (x - \mu)/\sigma\). A z-score of 2 means the value is two standard deviations above the mean, and -1 one standard deviation below.

Percentages and likelihoods can be read from a z-table using the z-score, but the z-score itself is a distance in standard deviations, not a percentage or a probability, and not an average of deviations.