Statistics: common misconceptions

If your child says… “The same average means equally consistent values.”

Ask together

Are the extreme dots equally far apart in both sets?

What’s true

An average does not tell us how far apart the values are.

If your child says… “The differences cancel, so there is no spread.”

Ask together

Do all the dots sit at the centre?

What’s true

Distances cannot cancel because each distance is non-negative.

If your child says… “There are two rows, so I divide by two.”

Ask together

How many observations do the two frequencies count altogether?

What’s true

The divisor counts observations, including repetitions.

If your child says… “Distance bars work like ordinary brackets.”

Ask together

Why does adding the signed differences first give a different answer?

What’s true

Taking absolute values before adding can differ from taking an absolute value after adding.

If your child says… “The standard deviation also uses square centimetres.”

Ask together

What happens to square centimetres when we take the square root?

What’s true

Taking the square root restores the measurements' original unit.

If your child says… “The squared distance for 3 only counts once.”

Ask together

How many readings of 3 appear in the expanded list?

What’s true

Each occurrence contributes, so repeated values require repeated squared distances.

If your child says… “I must add the assumed centre back to the variance too.”

Ask together

If a reading and its mean both rise equally, does their difference change?

What’s true

Adding the same number to every reading leaves spread unchanged. Multiplication changes standard deviation by the factor's absolute value, and variance by the factor squared.

If your child says… “I'll subtract 5 and add 3 in both totals.”

Ask together

Did the recorded 5 contribute 5 or its square to the square total?

What’s true

A square total needs squared contributions removed and replaced.