The normal distribution

A customer service centre employs 400 agents.
Their mean monthly salary is $\displaystyle {\mu}={9000}$ ₪.
The standard deviation is $\displaystyle {\sigma}={1000}$ ₪.

In grade 11 you learnt that the standard deviation measures how far the values spread around the mean.

Most agents earn close to 9000 ₪, and very few earn much more or much less. A drawing of all their salaries looks like a bell:

A distribution like this is called a normal distribution. In a normal distribution:

• The curve is symmetric about the mean: half the values are above it, and half below.
• The curve is highest at the mean: that is where most of the values are.
• The further from the mean, the fewer the values.
• The standard deviation sets the width: the greater $\displaystyle \sigma$, the wider and lower the bell.

Economic data about many people or many days is often spread normally: salaries, daily sales, delivery times, the weight a machine puts into each pack.

In every normal distribution, the area under the curve is split by the standard deviations in the same percents:

• Between $\displaystyle \mu$ and $\displaystyle \mu+\sigma$ lie 34% of the values, and so between $\displaystyle \mu-\sigma$ and $\displaystyle \mu$.
• Between $\displaystyle \mu+\sigma$ and $\displaystyle \mu+2\sigma$ lie 13.5%, and so between $\displaystyle \mu-2\sigma$ and $\displaystyle \mu-\sigma$.
• Above $\displaystyle \mu+2\sigma$ lie 2.5%, and so below $\displaystyle \mu-2\sigma$.

Adding up the bands gives:

• Within one standard deviation of the mean, between $\displaystyle \mu-\sigma$ and $\displaystyle \mu+\sigma$: $\displaystyle {34\%+34\%}={68\%}$ of the values.
• Within two standard deviations, between $\displaystyle \mu-2\sigma$ and $\displaystyle \mu+2\sigma$: $\displaystyle {13.5\%+68\%+13.5\%}={95\%}$.
• Above the mean: $\displaystyle {34\%+13.5\%+2.5\%}={50\%}$.

At the service centre: 68% of the agents earn between 8000 ₪ and 10,000 ₪, and 95% between 7000 ₪ and 11,000 ₪.

What percent of the agents earn more than 11,000 ₪?

The example below finds where 11,000 ₪ is on the axis, and adds up the bands above it:

• $\displaystyle {11000}={9000+2\cdot1000}$, that is $\displaystyle \mu+2\sigma$.
• Above $\displaystyle \mu+2\sigma$ there is one band: 2.5%.

And those who earn more than 8000 ₪? $\displaystyle {8000}={\mu-\sigma}$, and above it lie the band of 34% up to the mean, and the half above the mean:

The percent gives the number: the percent times the number of all the values.

The example below finds how many of the 400 agents earn between 9000 ₪ and 11,000 ₪:

• Between $\displaystyle \mu$ and $\displaystyle \mu+2\sigma$ there are two bands: $\displaystyle {34\%+13.5\%}={47.5\%}$.
• Out of 400 agents: $\displaystyle {0.475\cdot400}={190}$ agents.

A falafel stall’s daily takings are spread normally.
The mean takings are 4000 ₪ a day, and the standard deviation 500 ₪.
The stall works 200 days a year.

On what percent of the days are the takings between 4000 ₪ and 5000 ₪?

On what percent of the days are the takings below 3000 ₪?

On how many of the 200 days are the takings above 3500 ₪?

A food delivery company’s delivery time is spread normally.
The mean time is 40 minutes, and the standard deviation 8 minutes.

What percent of the deliveries take more than 48 minutes?

The company promises: “Your order within 56 minutes”.
What percent of the deliveries are late?

One evening 1000 deliveries went out.
About how many of them took between 32 and 56 minutes?

In two companies the salaries are spread normally, and the mean salary in both is 9000 ₪.
In company A the standard deviation is 1000 ₪, and in company B 2000 ₪.
The shaded area under each curve: the salaries above 11,000 ₪.

What percent of the workers in company A earn more than 11,000 ₪?

What percent of the workers in company B earn more than 11,000 ₪?

The mean is the same in the two companies.
Why does company B have more workers with a high salary?
And which company has more workers earning less than 7000 ₪?