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IB DP Maths AA SL Study Notes

4.5.2 Normal Distribution

Properties

The normal distribution is characterised by several distinctive properties:

  • Symmetry: One of the most recognisable features of the normal distribution is its symmetry. The curve is symmetric about the vertical line that passes through the mean. This means that the left half of the distribution is a mirror image of the right half.
  • Mean, Median, Mode: All three of these measures of central tendency are equal in a normal distribution and are located at the very centre of the distribution. For more detailed explanations on measures of central tendency, see Measures of Central Tendency.
  • Spread: The spread of data in a normal distribution follows a specific pattern. Approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and a whopping 99.7% within three standard deviations. This distribution is often referred to as the 68-95-99.7 rule. Understanding the Expected Value and Variance is crucial for comprehending how these percentages are calculated.
  • Asymptotic Nature: The curve of the normal distribution approaches, but never actually touches, the x-axis. This means that the tails of a normal distribution technically extend to infinity in both directions, which is a key characteristic of Continuous Random Variables.
  • Bell Shape: The curve's unique symmetrical shape has earned it the nickname "bell curve". This shape is what distinguishes the normal distribution from other types of distributions, such as the Binomial Distribution.
IB Maths Tutor Tip: Understanding the normal distribution's properties enhances your ability to analyse real-world data, making it pivotal for interpreting statistical information across various disciplines effectively.

Z-scores

Z-scores play a pivotal role in the realm of the normal distribution. Essentially, a z-score is a measure that describes a value's relationship to the mean of a group of values. It's quantified in terms of standard deviations from the mean. The formula for calculating the z-score of a value x is:

Z = (X - μ) / σ

Where:

  • X is the specific value.
  • μ is the mean of the distribution.
  • σ is the standard deviation.

The z-score provides insight into how many standard deviations a particular value is from the mean. A z-score of 0 indicates that the data point's score is identical to the mean score. For a deeper understanding of how z-scores fit within the broader context of normal distribution, consider exploring other aspects of the Normal Distribution.

Applications

The normal distribution is not just a theoretical concept; it has a plethora of real-world applications:

  • Quality Control: Many businesses harness the power of the normal distribution to monitor the quality of their products. For instance, if a product's dimensions or weight deviate too much from the mean (beyond a certain number of standard deviations), it might be considered defective.
  • Finance: The world of finance often assumes that stock returns follow a normal distribution. This assumption is harnessed in various models to value stock options and predict stock prices.
  • Medicine: Medical researchers might employ the normal distribution to analyse the data of a particular health metric from a sample population.
  • Education: In the rrealm of education, teachers might use the normal distribution to determine the grades of students in a large class. This application is closely related to understanding how various statistical measures interplay, such as the Correlation Coefficient.
IB Tutor Advice: Practise calculating and interpreting z-scores for diverse datasets to strengthen your grasp on normal distribution, crucial for excelling in questions about statistical analysis and probability in exams.

Example Question

Imagine the scores of students in a national exam are normally distributed with a mean of 50 and a standard deviation of 10. What is the z-score for a student who scored 70?

Answer: Using the z-score formula: Z = (X - μ) / σ Z = (70 - 50) / 10 Z = 2

This means the student's score is 2 standard deviations above the mean.

This means the student's score is 2 standard deviations above the mean. To explore more about how these calculations are applied in various contexts, reviewing Continuous Random Variables might provide additional insights.

FAQ

The 68-95-99.7 rule, also known as the empirical rule, provides a quick way to understand the spread of data in a normal distribution. It states that approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. This rule is significant because it gives a quick visual understanding of the distribution of data without having to delve into detailed calculations. It's a handy tool for making predictions and understanding the likelihood of certain outcomes.

The standard deviation, denoted as σ, determines the spread or width of the normal distribution. If the standard deviation is small, the curve will be narrow and tall, indicating that most of the data is clustered closely around the mean. Conversely, a large standard deviation results in a wider and flatter curve, indicating that the data is spread out over a larger range of values. In essence, the standard deviation provides a measure of the variability or dispersion of the data points from the mean.

In the real world, data is rarely, if ever, perfectly normally distributed. However, many datasets closely approximate the normal distribution, especially when the sample size is large. It's also worth noting that many statistical techniques are based on the assumption of normality, so even if data isn't perfectly normal, if it's "close enough," these techniques can still be applied. In cases where the assumption of normality is crucial, various tests, like the Shapiro-Wilk test, can be used to check for normality.

Yes, there are limitations. One primary limitation is the assumption of normality. Not all datasets follow a normal distribution, and applying techniques that assume normality to non-normal data can lead to incorrect conclusions. Additionally, the normal distribution is symmetric, which means it might not be suitable for data that is skewed. In such cases, other distributions like the log-normal or exponential might be more appropriate. It's essential to understand the underlying distribution of your data and the assumptions of any statistical techniques you're using.

The normal distribution is prevalent in statistics because of the Central Limit Theorem (CLT). The CLT states that the sum of a large number of independent and identically distributed random variables will be approximately normally distributed, regardless of the original distribution of the variables. This means that many processes, especially those that are the result of many small effects acting additively and independently, will be normally distributed. Additionally, the mathematical properties of the normal distribution make it tractable and easy to work with, further contributing to its widespread use in various fields.

Practice Questions

The heights of students in a school are normally distributed with a mean height of 165 cm and a standard deviation of 8 cm. What is the probability that a randomly selected student has a height between 150 cm and 180 cm?

To find the probability, we first need to calculate the z-scores for both 150 cm and 180 cm.

For 150 cm: Z = (150 - 165) / 8 = -1.875

For 180 cm: Z = (180 - 165) / 8 = 1.875

Using standard normal distribution tables or calculators, the probability for z = -1.875 is approximately 0.0307 and for z = 1.875 is approximately 0.9693.

The probability that a student's height is between 150 cm and 180 cm is 0.9693 - 0.0307 = 0.9386 or 93.86%.

A factory produces bolts, and the diameter of these bolts follows a normal distribution with a mean of 5 mm and a standard deviation of 0.2 mm. If the factory produces a bolt with a diameter less than 4.7 mm, it is considered defective. What percentage of bolts produced by the factory are defective?

First, we need to find the z-score for a bolt with a diameter of 4.7 mm.

Z = (4.7 - 5) / 0.2 = -1.5

Using standard normal distribution tables or calculators, the probability for z = -1.5 is approximately 0.0668.

This means that approximately 6.68% of the bolts produced by the factory are defective.

Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
LinkedIn
Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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