Statistics in the Punjab Police SI paper is narrow and very predictable. All 19 questions asked since 2016 are about three numbers — the mean, the median and the mode — and about one relationship between them. That relationship, Mode = 3 × Median − 2 × Mean, carries six questions on its own. Five more are plain mode-spotting, four need a median found by counting, and four read a frequency table. Nothing here requires standard deviation, variance or any formula beyond those on this page.
19
SI questions
2016–2023
6
use the empirical relation
Mode = 3Med − 2Mean
5
on the mode
count, don't compute
4
read a frequency table
grouped mean and median
Three averages, three jobs
Know what each one measures before you compute it.
QUICK CHECK
Compute the mean of the following data. 32, 27, 46, 12, 36, 81, 72, 27, 54, 43, 6, 92 (SI 2 Jul 2023)
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Answer: 44
Add all twelve values: 32 + 27 + 46 + 12 + 36 + 81 + 72 + 27 + 54 + 43 + 6 + 92 = 528. The mean is 528 ÷ 12 = 44.
Add in pairs that make round numbers — 32 + 46 + 12 = 90, 81 + 72 + 27 = 180 — rather than running down the list one at a time.
QUICK CHECK
What will be the mode of the following data? : 9, 7, 18, 16, 9, 16, 27, 18, 33, 9, 7, 16, 21, 18, 15, 7, 9 (SI 2 Jul 2023)
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Answer: 9
Tally the values rather than scanning. 9 appears four times; 7, 16 and 18 appear three times each; the rest once. The mode is 9.
With seventeen numbers it is worth writing a small tally column — a quick glance makes 16 or 18 look just as frequent.
QUICK CHECK
A researcher studying the behaviour of ants recorded the time (in seconds) taken by each ant to locate its food by considering 15 different ants as 51, 73, 63, 48, 69, 53, 37, 39, 44, 38, 42, 33, 72, 88, 25. The median time (in seconds) that the ants spent in searching their food is : (SI 29 Jun 2023)
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Answer: 48
Sort the 15 times: 25, 33, 37, 38, 39, 42, 44, 48, 51, 53, 63, 69, 72, 73, 88. With an odd count the median is the (15 + 1)/2 = 8th value, which is 48 seconds.
Never average the values — the mean here is 51.7, which is offered as option D.
QUICK CHECK
The median of observations 10, 12, 24, 28, x + 5, x + 9, 45, 47, 52, (x + 22) arranged in ascending order is 42. The value of the last observation of the data is : (SI 9 Jul 2023)
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Answer: 57
Ten observations, so the median is the average of the 5th and 6th: ((x + 5) + (x + 9))/2 = x + 7. Setting x + 7 = 42 gives x = 35.
The last observation is x + 22 = 35 + 22 = 57. The question asks for the last value, not for x — 35 itself is not on offer, but 53 and 60 are there for near misses.
QUICK CHECK
The integer ‘x’ is the median of seven different integers 23, 19, 16, 21, 25, 17 and x. Find the value of ‘x’. (SI 14 Jul 2023)
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Answer: 20
Sort the six known integers: 16, 17, 19, 21, 23, 25. With x added there are seven values, so the median is the 4th. For x itself to be the 4th it must sit between 19 and 21, and since all seven integers are different, x = 20.
Check: 16, 17, 19, 20, 21, 23, 25 — the 4th value is 20 ✓.
QUICK CHECK
If m (a whole number) is the median of the numbers 28, 13, 18, 26, 10, m, 32, then which of the following cannot be a possible value of m? (SI 20 Jun 2023)
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Answer: 27
Sort the six known numbers: 10, 13, 18, 26, 28, 32. With m there are seven values and m is stated to be the median, so m must land in the 4th position — that is, between 18 and 26.
Among the options 23, 25 and 19 all satisfy that, but 27 is larger than 26, so it would sit 5th and the median would be 26 instead. Hence 27 cannot be a value of m.
QUICK CHECK
For the data 20, 36, 13, (x + 3), 31, 35, (x − 4), 31, 36, 4, 13, 20, 5, if the mean is 23, then what will be the mode of the data? (SI 14 Jul 2023)
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Answer: 31
Thirteen values with mean 23 means the total is 13 × 23 = 299. The eleven plain numbers add to 20 + 36 + 13 + 31 + 35 + 31 + 36 + 4 + 13 + 20 + 5 = 244, and the two algebraic ones add to (x + 3) + (x − 4) = 2x − 1. So 244 + 2x − 1 = 299, giving 2x = 56 and x = 28.
The data is now 20, 36, 13, 31, 31, 35, 24, 31, 36, 4, 13, 20, 5. 31 appears three times, more than any other value, so the mode is 31.
QUICK CHECK
If the mode of the data 78, 74, 62, 2x – 3, 57, 62, 57, 48 is 57, then what is the value of (x + 3)? (SI 29 Jul 2023)
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Answer: 33
In 78, 74, 62, 2x − 3, 57, 62, 57, 48 the value 57 already appears twice and so does 62. For 57 to be the mode it must appear more often than 62, so the unknown entry must also be 57: 2x − 3 = 57 gives x = 30.
The question asks for x + 3 = 33, not for x. Option C (30) is x itself, placed there for exactly this slip.
CHECK YOURSELF
Find the mode of the following data. 15, 17, 26, 28, 16, 10, 28, 16, 28 and 16 (SI 15 Jul 2023)
Show answer
Count carefully: in 15, 17, 26, 28, 16, 10, 28, 16, 28, 16 the value 16 appears three times and 28 also appears three times. The data as printed is genuinely bimodal, which a single-answer MCQ cannot express — the paper's key is 16, so mark that, but note that 28 is equally frequent and that this is a defect in the question rather than in your counting. If a mode question ever gives you two values tied at the top, recount once, then take the one that appears among the options.
The empirical relation
Six questions from a single line.
KEY TERM
Mode = 3 × Median − 2 × Mean
For a moderately skewed distribution the three averages sit roughly in line, with the median one-third of the way from the mean to the mode. Rearranged, the same line gives you the other two: Median = (Mode + 2 × Mean) ÷ 3 · Mean = (3 × Median − Mode) ÷ 2 You only ever need to remember the first form; the other two follow in one step of algebra. Every SI question on this topic gives you two of the three and asks for the third.
Mode = 3Med − 2Mean→rearrange→any two give the third
QUICK CHECK
The mode of a distribution is given as 8.1 and the median of the distribution is given as 9.9. Using the empirical relationship between arithmetic mean, median and mode, find the arithmetic mean of the given distribution. (SI 15 Jul 2023)
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Answer: 10.8
Substitute directly: Mode = 3 × Median − 2 × Mean, so 8.1 = 3(9.9) − 2 × Mean = 29.7 − 2 × Mean. Then 2 × Mean = 21.6 and the mean is 10.8.
A sanity check: mode 8.1 < median 9.9, so the distribution leans right and the mean must be the largest of the three — 10.8 fits.
QUICK CHECK
If the values of the mean and the mode are 10 and 7, respectively, find the value of the median by using the empirical relation between mean, median and mode. (SI 29 Jun 2023)
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Answer: 9
Mode = 3 × Median − 2 × Mean gives 7 = 3 × Median − 20, so 3 × Median = 27 and the median is 9.
Here mode 7 < mean 10, so the median must lie between them — and 9 does.
QUICK CHECK
In an asymmetrical distribution, if the mean and median of the distribution are 590 and 470, respectively, then the mode is : (SI 20 Jun 2023)
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Answer: 230
Mode = 3 × 470 − 2 × 590 = 1410 − 1180 = 230.
This distribution is strongly left-skewed: the mean 590 is far above the median 470, so the mode is dragged well below both. A negative or very small mode is normal in such cases and is not a sign of an arithmetic slip.
QUICK CHECK
The mean of the data using the relation between mean, median and mode when it is given that mode is 32.5 and median is 25.5, is : (SI 31 Jul 2023)
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Answer: 22
Mode = 3 × Median − 2 × Mean gives 32.5 = 3(25.5) − 2 × Mean = 76.5 − 2 × Mean. So 2 × Mean = 44 and the mean is 22.
Option D (44) is 2 × Mean, left one step short — always finish the division.
QUICK CHECK
For a certain data set, the mode is 2 more than the median and its mean is 12. Find the median of the data. (SI 20 Jun 2023)
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Answer: 13
Write the unknown median as M. The mode is M + 2 and the mean is 12, so the relation gives M + 2 = 3M − 24.
That rearranges to 2M = 26, so the median is 13 (and the mode is 15).
When the question states a relationship instead of a value, substitute it into the formula and solve — there is no second equation to find.
QUICK CHECK
The mode of certain data is 6 more than its mean, and the median of this data is 78. What will the mode of the data be? (SI 29 Jun 2023)
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Answer: 82
Let the mean be m, so the mode is m + 6. With the median at 78 the relation gives m + 6 = 3(78) − 2m = 234 − 2m.
So 3m = 228 and m = 76, making the mode 76 + 6 = 82. The question asks for the mode, so 76 — the mean — is the trap, and it is offered as option A.
Reading a frequency table
Four questions, and the same two traps every time.
THE TWO MISTAKES A TABLE INVITES
First, giving the frequency instead of the value. The mode of a frequency table is the observation with the highest frequency, not the frequency itself. If the mark 15 was scored by 16 students and that is the biggest count, the mode is 15, not 16. Second, averaging the row of values and ignoring the frequencies entirely. That only happens to work when the frequencies are symmetric, which examiners deliberately arrange in one or two questions so that the lazy method appears to succeed.
QUICK CHECK
The number of marks obtained by 60 students of a class are given in the table. Find the mode. (SI 2 Jul 2023)
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Answer: 15
The frequencies in the table are 7, 9, 16, 10, 10 and 8, and they add to 60 — which matches the 60 students, so the table has been read correctly.
The largest frequency is 16, and it belongs to the mark 15. Option D's 16 is the frequency, not the mark; option A's 53 is the highest mark, which is the maximum rather than the mode.
QUICK CHECK
Find the mode of the following data. (SI 29 Jul 2023)
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Answer: 78
The frequencies in the table are 7, 3, 12, 10 and 5. The largest is 12, and it belongs to the variable 78.
Option B (108) is the largest variable in the table — the maximum, not the mode. Option D (80) is the first value listed. Both are there to catch a glance rather than a read.
QUICK CHECK
An apartment has 85 flats. The frequency distribution given below shows the monthly consumption of electricity. What will be the average consumption per flat? (SI 20 Jun 2023)
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Answer: 99.8235
For grouped data the mean is Σfx ÷ Σf, where x is each class mid-point. The mid-points are 58, 73, 88, 103, 118, 133 and 148, with frequencies 11, 8, 14, 22, 18, 9 and 3.
Σfx = 638 + 584 + 1232 + 2266 + 2124 + 1197 + 444 = 8485, and Σf = 85, which confirms the 85 flats. The mean is 8485 ÷ 85 = 99.8235.
A quick check before dividing: the heaviest class is 96–110, so the answer must sit a little below its mid-point 103 because of the long lower tail — which already rules out 102.5315 and 101.9328.
QUICK CHECK
The following table shows weights of 60 students in a class. Find the median weight of the students. (SI 29 Jul 2023)
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Answer: 40 kg
With n = 60 the median is the average of the 30th and 31st values in order, so build the running total: 10 kg → 4, 20 kg → 11, 30 kg → 23, 40 kg → 38, 50 kg → 51, 60 kg → 56, 70 kg → 60.
The running total jumps from 23 to 38 at the 40 kg column, so the 24th through 38th students all weigh 40 kg — including the 30th and the 31st. The median is 40 kg.
Averaging the seven weights, (10 + 20 + … + 70) ÷ 7, also gives 40 here, but only because these frequencies happen to be near-symmetric. On a lopsided table that shortcut fails.
COMMON TRAPS
- Giving the frequency when the question asked for the observation. The mode of a table is the value in the top row, not the count below it.
- Taking the median without sorting first. The data in the stem is almost never in order.
- Stopping at 2 × Mean or at x instead of finishing the last step. Several SI options are exactly the half-done value.
- Mixing up the empirical relation. It is Mode = 3 Median − 2 Mean, with the 3 on the median — never 3 Mean − 2 Median.
- Averaging a frequency table's value row and ignoring the frequencies. It works only when the frequencies are symmetric.
QUICK BYTES
- Mode = 3 × Median − 2 × Mean
- Median = (Mode + 2 × Mean) ÷ 3
- Mean = (3 × Median − Mode) ÷ 2
- Odd n: median is the (n + 1)/2-th value
- Even n: average the two middle values
- Grouped mean = Σfx ÷ Σf, x = class mid-point
- Table median: use the running total
- The mode is a value, never a frequency
REMEMBER
Mo-Me-Mn, three-two. Write the three averages in that order — Mode, Median, Mean — and the formula reads straight off it: the first equals three times the second minus two times the third. The coefficients 3 and 2 go with the two words you did not start from.