# The normal distribution · Grade 12 · 3 units

> The normal distribution, Grade 12 · 3 units: Recognising the normal curve and more. Explanation, worked examples and free practice following the Israeli curriculum.

In The normal distribution (Grade 12 · 3 units) you practise 8 skills: Recognising the normal curve · Reading the chart · The share with a property · The value from a share · Using symmetry · The mean and the standard deviation from shares · The standard score · Comparing two populations.

In a normal distribution the curve is a bell, symmetric about the mean: its peak is above the mean, and the mean, the median and the mode are equal. The larger the standard deviation, the wider and lower the bell and the more spread out the values. A curve with two peaks, a flat top or a long tail on one side is not normal.

On the chart of the distribution, the axis marks the mean and the values half a standard deviation apart from it, on both sides. Each band between two neighbouring marks shows, by the chart, the percentage of the values in it: 19.1 percent in each of the two bands next to the mean, then 15, 9.2, 4.4, 1.7 and 0.5. Add bands to find the share above a value, below it or between two values, and from a given share find the value, the mean or the standard deviation.

The standard score of a value is its difference from the mean, divided by the standard deviation. It says how many standard deviations the value lies from the mean, and on which side: a positive score is a value above the mean, and a negative score a value below it. Standard scores compare values from two different populations: the value with the higher standard score stands higher compared with its own population.

**Tip:** In every question, first mark the region asked for on the chart, and only then add up its bands.

## What you learn

### Recognising the normal curve

- Example: Four curves are given. Curve A: (Curve A: a curve that rises to a long flat top and then falls) Curve B: (Curve B: a curve with two peaks and a dip between them) Curve C: (Curve C: a curve whose peak is near the left, falling slowly towards the right) Curve D: (Curve D: a curve with one rounded peak in the middle that falls gently towards both ends) Which curve shows a normal distribution?
- Options: Curve A · Curve B · Curve C · Curve D
- Answer: Curve D
- Solution: Curve D has one rounded peak in the middle and is symmetric about it, so it shows a normal distribution. Of the others, one has a flat top, one has two peaks and one is not symmetric.

### Reading the chart

- Example: The marks in a national test are normally distributed with mean 74 and standard deviation 4. The chart shows the percentage of the values in each band of half a standard deviation. (The normal distribution chart with mean 74 and standard deviation 4: 0.1 percent below 62; 0.5 percent between 62 and 64; 1.7 percent between 64 and 66; 4.4 percent between 66 and 68; 9.2 percent between 68 and 70; 15 percent between 70 and 72; 19.1 percent between 72 and 74; 19.1 percent between 74 and 76; 15 percent between 76 and 78; 9.2 percent between 78 and 80; 4.4 percent between 80 and 82; 1.7 percent between 82 and 84; 0.5 percent between 84 and 86; and 0.1 percent above 86.) What value lies two and a half standard deviations above the mean?
- Answer: 84
- Solution: Each mark on the axis is half a standard deviation from the next. The value asked for is 74 + 2.5 × 4 = 84 , that is 84.

### The share with a property

- Example: The hourly pay of the workers in a company is normally distributed with mean 45 shekels and standard deviation 4 shekels. (The normal distribution chart with mean 45 and standard deviation 4: 0.1 percent below 33; 0.5 percent between 33 and 35; 1.7 percent between 35 and 37; 4.4 percent between 37 and 39; 9.2 percent between 39 and 41; 15 percent between 41 and 43; 19.1 percent between 43 and 45; 19.1 percent between 45 and 47; 15 percent between 47 and 49; 9.2 percent between 49 and 51; 4.4 percent between 51 and 53; 1.7 percent between 53 and 55; 0.5 percent between 55 and 57; and 0.1 percent above 57.) What percent of the workers have an hourly pay below 37 shekels?
- Answer: 2.3%
- Solution: By the chart, below 37 lie the tail 0.1 and the bands 0.5 and 1.7, together 2.3 percent.

### The value from a share

- Example: The marks in a national test are normally distributed with mean 74 and standard deviation 4. (The normal distribution chart with mean 74 and standard deviation 4: 0.1 percent below 62; 0.5 percent between 62 and 64; 1.7 percent between 64 and 66; 4.4 percent between 66 and 68; 9.2 percent between 68 and 70; 15 percent between 70 and 72; 19.1 percent between 72 and 74; 19.1 percent between 74 and 76; 15 percent between 76 and 78; 9.2 percent between 78 and 80; 4.4 percent between 80 and 82; 1.7 percent between 82 and 84; 0.5 percent between 84 and 86; and 0.1 percent above 86.) About 98 percent of the pupils have a mark above a certain value. What is that value?
- Answer: 66
- Solution: Above the value two standard deviations below the mean lie about 98 percent (97.7 percent by the chart). So the value is 74 − 2 × 4 = 66 , that is 66.

### Using symmetry

- Example: The hourly pay of the workers in a company is normally distributed, and its mode is 59 shekels. A third of the workers have an hourly pay above 60 shekels. What fraction of the workers have an hourly pay between 58 shekels and 60 shekels?
- Answer: 1/3
- Solution: The values 58 shekels and 60 shekels are the same distance, 1 shekel, from the mode 59 shekels, which is also the mean, so the share below 58 shekels is the same as the share above 60 shekels. Above 60 shekels lies a third and below 58 shekels lies a third, so a third is left between them: 1/3 .

### The mean and the standard deviation from shares

- Example: The marks in a national test are normally distributed with mean 74. It is known that about 7 percent of the pupils have a mark above 80. What is the standard deviation?
- Answer: 4
- Solution: About 7 percent (6.7 percent by the chart) lie above the value one and a half standard deviations above the mean, so 80 lies one and a half standard deviations above the mean. So 1.5 × σ = 80 − 74 = 6 , which gives σ = 4 .

### The standard score

- Example: The marks in a national test are normally distributed with mean 74 and standard deviation 4. (The normal distribution chart with mean 74 and standard deviation 4: 0.1 percent below 62; 0.5 percent between 62 and 64; 1.7 percent between 64 and 66; 4.4 percent between 66 and 68; 9.2 percent between 68 and 70; 15 percent between 70 and 72; 19.1 percent between 72 and 74; 19.1 percent between 74 and 76; 15 percent between 76 and 78; 9.2 percent between 78 and 80; 4.4 percent between 80 and 82; 1.7 percent between 82 and 84; 0.5 percent between 84 and 86; and 0.1 percent above 86.) What is the standard score of the mark 72?
- Answer: −0.5
- Solution: By the formula z = (72 − 74) ÷ 4 = −0.5 .

### Comparing two populations

- Example: In the northern town the heights of the adults are normally distributed with mean 182 cm and standard deviation 10 cm, and in the southern town with mean 164 cm and standard deviation 4 cm. Noa, from the northern town, is 169 cm tall, and Maya, from the southern town, is 168 cm tall. Who is relatively taller, compared with their own town?
- Options: Noa · Maya
- Answer: Maya
- Solution: The standard score of Noa: z = (169 − 182) ÷ 10 = −1.3 . The standard score of Maya: z = (168 − 164) ÷ 4 = 1 . The standard score of Maya is higher, so the answer is Maya.

## Not on the site yet

- The normal table: values off the chart
- Explaining the standard deviation in words
- Calculations for distributions that are not normal

The exercises are made by rules and the answers are worked out in code. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 3 units (grades 10 and 11).

[Practise on the site](https://kixplo.com/il/en/) · [All Grade 12 · 3 units topics](https://kixplo.com/il/en/math/grade-12-3u/)