Binomial and Normal Distribution Characteristics

Two famous distributions

A distribution describes how the values of a variable are spread out. Two of the most useful are the binomial (for counting successes) and the normal (a symmetrical bell curve). This module drills what makes each one distinctive, when to use it, and how to find the mean of a binomial distribution.

Distribution words

Four ideas run through the topic:

Match each term to its characteristic

Which is binomial?

Which situation is best modelled by a binomial distribution?

• The number of heads in 20 coin flips
• The heights of students in a year group
• The exact finishing time of a race
• The temperature measured through a day

When is it binomial?

A distribution is binomial when four conditions hold: 1. A fixed number of trials, n 2. Each trial has only two outcomes (success or failure) 3. The trials are independent 4. The probability of success, p, is constant For a binomial distribution, the mean = n x p.

Binomial conditions

Select the TWO statements that are conditions for a binomial distribution.

• There is a fixed number of trials
• Each trial has only two outcomes
• The trials affect one another
• The probability changes at each trial

Binomial versus normal

The two distributions suit different kinds of data.

The mean of a binomial

A binomial distribution has n = 20 trials and a probability of success p = 0.5. What is its mean? (Mean = n x p.)

The shape of the normal

In a perfectly normal distribution, the mean, median and mode are:

• All equal, at the centre of the curve
• All different from each other
• Found at the two tails of the curve
• Impossible to find

One more mean

A binomial distribution has n = 10 trials and a probability of success p = 0.3. What is its mean? (Mean = n x p.)

Put it together

A _____ distribution counts successes in a fixed number of trials and is _____. Its mean is n times _____. A _____ distribution is a symmetrical bell curve for _____ data, where the mean, median and mode are equal.

Choose a distribution

Put the steps of choosing which distribution to use into a sensible order.

• Look at the data: is it counted or measured?
• If counted, with fixed trials and two outcomes, consider the binomial
• If measured and symmetrical, consider the normal
• Check that the conditions actually match
• Use the distribution to describe or predict

Which distribution?

Three sets of data. Choose the distribution that best fits each one.

• A teacher records the number of correct answers out of 10 for each pupil in a quiz. Which distribution fits?
• A nurse records the birth weights of many babies; they cluster symmetrically around an average. Which distribution fits?
• You need the mean of a binomial with n = 40 trials and p = 0.25. What is the mean?

Match the distribution

Top answers match the distribution to the data: a count of successes over a fixed number of trials is binomial; a symmetrical, continuous measurement is normal. Justify the choice from the characteristics (discrete vs continuous, fixed trials, symmetry), and remember the binomial mean is simply n x p. Identify, justify, calculate - top band.