Q.What is a symmetric distribution? Distinguish it from a skewed distribution, and state the relationship among mean, median and mode in each case.

A symmetric distribution has its two halves mirroring each other around the centre, while a skewed distribution has one tail longer than the other.

Symmetric distribution. A frequency distribution is symmetric when its frequency curve is a mirror image about the central value — for every observation a fixed distance below the centre, there is an equally frequent observation the same distance above it. In this case the three measures of central tendency coincide:

and the two tails of the distribution (above and below the centre) are equal in length and shape.

Skewed distribution. A frequency distribution is skewed (asymmetrical) when this mirror-image property breaks down — one tail is longer or fatter than the other, so the bulk of the data is concentrated toward one end while a smaller number of extreme values stretch out the opposite tail. Because the mean is pulled toward the long tail far more than the mode is, the three averages separate:

• Positively skewed: long tail on the right (higher values) → Mean > Median > Mode.
• Negatively skewed: long tail on the left (lower values) → Mean < Median < Mode.

Why this matters. Two distributions can have exactly the same mean and the same standard deviation and still look very different once plotted — one a neat symmetric bell, the other badly lopsided with a long tail on one side. Skewness captures precisely this shape difference, which central tendency and dispersion alone cannot reveal.

A common error is assuming that 'Mean = Median = Mode' is only a definition to memorise, without checking it against the actual shape of the curve — always connect the numeric condition (equal averages) to the visual condition (equal, mirror-image tails). Another frequent error is reversing the positive/negative skew inequality — remember the mean, being most sensitive to extreme values, is always the one pulled furthest toward the long tail, so it sits at the far end of the inequality on the side the tail points.