S1 · Normal Distribution and z-Scores

Alberta Grade 11 Mathematics

Statistics

Mathematics 20-2 — Study topic S1

What you will learn

• Describe the main features of a normal distribution.
• Interpret standard deviation as a measure of spread around a mean.
• Use the 68–95–99.7 rule to interpret approximate proportions in a normal distribution.
• Interpret a z-score as the number of standard deviations an observation is from the mean.

1. Review: mean, data, and units

• The mean describes the data's average value.
• Compare values only when they use the same units.

2. Recognizing a normal distribution

• A normal curve is symmetric and bell-shaped, with its mean at the centre.
• Standard deviation describes spread in the data's units.
• The 68–95–99.7 rule describes approximate proportions within one, two, and three standard deviations.

3. Interpreting a z-score

• The sign tells whether the observation is above or below the mean.
• The absolute value tells how many standard deviations away it is.
• A z-score has no units.

4. Read the value, then check its meaning

• State the meaning of a z-score in context, including above or below the mean.
• Use the normal-rule percentages as approximations.
• Check the sign and size against the observation's position.

Worked example

Interpreting the centre and spread

1. Identify the centre
   The mean is the centre of this normal distribution. A score of 72 points is the average, not a score that every student received.
   $$\mu=72\text{ points}$$
2. Interpret the spread
   The standard deviation is 6 points. Scores commonly differ from the mean by a distance measured in points; the standard deviation describes the distribution's spread.
   $$\sigma=6\text{ points}$$
3. Find the one-standard-deviation interval
   One standard deviation below the mean is 72 minus 6, and one above is 72 plus 6. About 68% of scores are expected in this interval if the normal model is suitable.
   $$72-6=66,\quad72+6=78$$

Worked example

Finding and interpreting a z-score

1. Compare with the mean
   The trip took 6 minutes longer than the mean. This difference is positive because the observation is above the mean.
   $$30-24=6\text{ minutes}$$
2. Measure the difference in standard deviations
   Divide the 6-minute difference by the standard deviation of 3 minutes. The units cancel, leaving a z-score with no unit.
   $$z=\frac{30-24}{3}=2$$
3. Interpret the result
   The positive sign means the trip time is above the mean. The value 2 means it is two standard deviations above the mean. In a normal distribution, that is farther from the centre than most values.
   $$z=2$$

Worked example

Interpreting a negative z-score

1. Find the difference from the mean
   The package is lighter than the mean, so subtracting the mean gives a negative difference.
   $$470-500=-30\text{ grams}$$
2. Calculate the z-score
   Divide the difference by the standard deviation. Both quantities are in grams, so the units cancel.
   $$z=\frac{470-500}{20}=-1.5$$
3. Connect the score to the normal model
   The package is 1.5 standard deviations below the mean. It lies outside the interval within one standard deviation, but inside the interval within two standard deviations. About 68% of values are within one standard deviation and about 95% within two, so the package is in the broader outer portion between those intervals, not beyond two standard deviations.
   $$500-2(20)=460,\quad 500-20=480$$

Common mistakes and how to avoid them

Lesson summary

• A normal distribution is approximately symmetric and bell-shaped, with the mean at its centre.
• Standard deviation describes spread and uses the same unit as the data.
• About 68%, 95%, and 99.7% of values lie within one, two, and three standard deviations of the mean, respectively.
• A z-score gives an observation's distance from the mean in standard-deviation units: positive is above, negative is below.

Check your understanding

Key terms

Distribution

A description of the values in data and how often they occur.

Normal distribution

A symmetric, bell-shaped model in which values near the mean are more common than values far from it.

Mean

The average of a set of values.

Standard deviation

A measure of how spread out values are around the mean.

z-score

A value that states how many standard deviations an observation is above or below the mean.

Continue through Math 20-2

View the complete Math 20-2 Alberta Grade 11 Mathematics curriculum and lessons

• NL4 · Solve square-root and cube-root equations and check proposed roots
• S2 · Interpret confidence intervals, confidence levels, and margins of error
• M1 · Apply and compare rates and unit rates in context
• M3 · Relate scale factors to area, surface area, and volume
• G1 · Derive direct proofs involving angles and triangles
• G2 · Solve problems using properties of angles and triangles

Tutor support for S1 · Interpret normal distributions, standard deviation, and z-scores

Working on S1 · Interpret normal distributions, standard deviation, and z-scores in Math 20-2? Bring the question and your own attempt. In a live session, you can identify the data and assumptions, choose a suitable representation, and interpret the result in context. Ask about the first step that does not make sense, then try a similar question yourself.

Explore math tutoring · Request an eligible free 30-minute introduction. Your request is matched by subject and language; a tutor must accept before a time is confirmed.

برای مبحث S1 · توزیع نرمال، انحراف معیار و نمرهٔ z را تفسیر کنید در درس Math 20-2، سؤال و تلاش خود را همراه بیاورید. در جلسه می‌توانید داده‌ها و فرض‌ها را مشخص کنید، نمایش مناسبی برگزینید و نتیجه را در متن مسئله تفسیر کنید. راهنمای معلم ریاضی را ببینید یا جلسهٔ معارفهٔ رایگان ۳۰ دقیقه‌ای درخواست کنید. زمان جلسه پس از پذیرش معلم قطعی می‌شود.

About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Alberta Grade 11 Mathematics (Math 20-2), expectation S1. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

Official curriculum reference

Report a correction or ask a question