Deviation Concept Quiz: Concept of Deviation

Deviation is calculated by subtracting the mean from the data value: deviation equals value minus mean. If the result is positive, the value is above the mean. If the result is negative, the value is below the mean. If the result is zero, the value equals the mean. For example if the mean is 10 and a value is 14, the deviation is 14 minus 10 equals 4.

Deviation equals value minus mean equals 4 minus 8 equals negative 4. The negative sign confirms that 4 is below the mean of 8. The magnitude of 4 tells us the value is exactly 4 units away from the mean. If the value were above the mean by the same amount, the deviation would be positive 4 instead.

The answer is True. Deviation equals value minus mean. When the value and the mean are the same number, subtracting gives zero. For example if the mean is 7 and a value is 7, the deviation is 7 minus 7 equals 0. A value sitting exactly at the mean has no distance from it in either direction, so its deviation is always zero.

Deviation equals value minus mean equals 9 minus 6 equals 3. The positive result confirms that 9 is above the mean of 6. The value 9 is exactly 3 units above the mean. If the value were 3 instead, the deviation would be 3 minus 6 equals negative 3, placing it 3 units below the mean.

Computing each deviation: 10 minus 20 equals negative 10, 20 minus 20 equals 0, and 30 minus 20 equals 10. The deviations are negative 10, 0, and 10. Notice that the sum of these deviations is negative 10 plus 0 plus 10 equals 0, which illustrates the rule that deviations from the mean always sum to zero.

Option A: 6 minus 9 equals negative 3 — this is a negative deviation. Option C: 7 minus 9 equals negative 2 — this is also a negative deviation. Option B: 11 minus 9 equals 2 — this is a positive deviation, not negative. Option D: 9 minus 9 equals 0 — this is a zero deviation, not negative. A negative deviation always means the value is below the mean.

Computing each absolute deviation: the absolute value of 3 minus 8 equals 5. The absolute value of 7 minus 8 equals 1. The absolute value of 14 minus 8 equals 6. The value 14 has the largest absolute deviation of 6 because it is furthest from the mean. Note that being the largest value in a dataset does not always mean having the largest deviation — what matters is distance from the mean.

The answer is False. Deviation is calculated as value minus mean, not mean minus value. Reversing the order would flip all signs, turning positive deviations negative and negative deviations positive. For example if the mean is 5 and a value is 8, the correct deviation is 8 minus 5 equals 3, not 5 minus 8 equals negative 3. The formula is always value minus mean.

Deviation equals value minus mean equals 11 minus 15 equals negative 4. The negative sign confirms that 11 is below the mean of 15. The value is exactly 4 units below the mean. Option A (4) has the correct magnitude but the wrong sign — since the value is below the mean, the deviation must be negative.

Computing each deviation: 2 minus 5 equals negative 3, 5 minus 5 equals 0, and 8 minus 5 equals 3. Adding them: negative 3 plus 0 plus 3 equals 0. This illustrates the fundamental rule that deviations from the mean always sum to exactly zero for any data set. The positive deviations above the mean and the negative deviations below the mean cancel each other perfectly.

Computing each deviation: 6 minus 9 equals negative 3, 8 minus 9 equals negative 1, 10 minus 9 equals 1, and 12 minus 9 equals 3. The deviations are negative 3, negative 1, 1, and 3. Confirming: the sum is negative 3 plus negative 1 plus 1 plus 3 equals 0, as expected. All four values are equally spaced from the mean in pairs.

A negative deviation means the value is below the mean. The magnitude of 7 tells us the value is exactly 7 units below the mean. For example if the mean is 10, a deviation of negative 7 means the value is 10 minus 7 equals 3. The sign of the deviation tells us direction (above or below the mean) and the size tells us the distance from the mean.

The answer is True. The absolute value of both deviations is 4, meaning both values are exactly 4 units away from the mean. One is 4 units above and the other is 4 units below, but their distance from the mean is equal. For example if the mean is 10, a deviation of positive 4 gives the value 14 and a deviation of negative 4 gives the value 6, both exactly 4 units from 10.

Computing each absolute deviation: the absolute value of 2 minus 7.5 equals 5.5. The absolute value of 6 minus 7.5 equals 1.5. The absolute value of 8 minus 7.5 equals 0.5. The absolute value of 14 minus 7.5 equals 6.5. The value 14 has the largest absolute deviation of 6.5 because it is furthest from the mean of 7.5.

Option A is true: positive and negative deviations always cancel exactly, making their sum zero for any dataset. Option C is true: a negative deviation results from a value that is smaller than the mean, placing it below the mean. Option B is false: deviations can be decimals or fractions whenever the mean is not a whole number. Option D is false: the largest deviation belongs to the value furthest from the mean, which is not necessarily the largest value.

Since all deviations from the mean must sum to zero, the missing deviation equals zero minus (negative 2 plus negative 1) equals zero minus negative 3 equals 3. The missing value is therefore 6 plus 3 equals 9. Confirming: negative 2 plus negative 1 plus 3 equals 0, which satisfies the rule that deviations always sum to zero.

Every value in this data set equals the mean of 4. Since deviation equals value minus mean, each deviation is 4 minus 4 equals 0. This data set has zero spread — all values are identical and sit exactly at the mean. When all deviations are 0, the data has no variability whatsoever.

The answer is False. If all deviations equal 0, then every value equals the mean. Since every value equals the same number, all values in the data set must be identical. It is impossible for values to differ from each other when all their deviations are 0. For example the only four-value data set where all deviations are 0 is one where all four values are the same number.

Computing each deviation: 3 minus 5 equals negative 2 (negative), 5 minus 5 equals 0 (zero), and 7 minus 5 equals 2 (positive). Only the value 7 has a positive deviation because it is the only value above the mean of 5. The value 3 has a negative deviation, the value 5 has a zero deviation, and not all values have positive deviations.

Equal and opposite deviations of negative 5 and positive 5, with one value exactly at the mean (deviation 0), indicate the data is perfectly symmetric around the mean. The value with deviation negative 5 is 5 units below the mean and the value with deviation positive 5 is 5 units above — mirroring each other exactly. This also confirms the deviations sum to zero: negative 5 plus 0 plus 5 equals 0.