Bell Shaped Curve: Normal Distribution In Statistics (2024)

Bell Shaped Curve: Normal Distribution In Statistics (1)

Properties of normal distribution

The normal distribution is a continuous probability distribution that is symmetrical on both sides of the mean, so the right side of the center is a mirror image of the left side.

The area under the normal distribution curve represents the probability and the total area under the curve sums to one.

Most of the continuous data values in a normal distribution tend to cluster around the mean, and the further a value is from the mean, the less likely it is to occur. The tails are asymptotic, which means that they approach but never quite meet the horizon (i.e., the x-axis).

For a perfectly normal distribution, the mean, median, and mode will be the same value, visually represented by the peak of the curve.

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The normal distribution is often called the bell curve because the graph of its probability density looks like a bell. It is also known as called Gaussian distribution, after the German mathematician Carl Gauss who first described it.

Normal distribution Vs. Standard normal distribution?

A normal distribution is determined by two parameters the mean and the variance. A normal distribution with a mean of 0 and a standard deviation of 1 is called a standard normal distribution.

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Figure 1. A standard normal distribution (SND).

This is the distribution that is used to construct tables of the normal distribution.

Why is the normal distribution important?

The bell-shaped curve is a common feature of nature and psychology

The normal distribution is the most important probability distribution in statistics because many continuous data in nature and psychology display this bell-shaped curve when compiled and graphed.

For example, if we randomly sampled 100 individuals, we would expect to see a normal distribution frequency curve for many continuous variables, such as IQ, height, weight, and blood pressure.

Parametric significance tests require a normal distribution of the sample’s data points

The most powerful (parametric) statistical tests psychologists use require data to be normally distributed. If the data does not resemble a bell curve, researchers may use a less powerful statistical test called non-parametric statistics.

Converting the raw scores of a normal distribution to z-scores

We can standardize a normal distribution’s values (raw scores) by converting them into z-scores.

This procedure allows researchers to determine the proportion of the values that fall within a specified number of standard deviations from the mean (i.e., calculate the empirical rule).

What is the empirical rule formula?

The empirical rule in statistics allows researchers to determine the proportion of values that fall within certain distances from the mean. The empirical rule is often referred to as the three-sigma rule or the 68-95-99.7 rule.

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If the data values in a normal distribution are converted to standard score (z-score) in a standard normal distribution, the empirical rule describes the percentage of the data that fall within specific numbers of standard deviations (σ) from the mean (μ) for bell-shaped curves.

The empirical rule allows researchers to calculate the probability of randomly obtaining a score from a normal distribution.

68% of data falls within the first standard deviation from the mean. This means there is a 68% probability of randomly selecting a score between -1 and +1 standard deviations from the mean.

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95% of the values fall within two standard deviations from the mean. This means there is a 95% probability of randomly selecting a score between -2 and +2 standard deviations from the mean.

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99.7% of data will fall within three standard deviations from the mean. This means there is a 99.7% probability of randomly selecting a score between -3 and +3 standard deviations from the mean.

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How to check data

Statistical software (such as SPSS) can be used to check if your dataset is normally distributed by calculating the three measures of central tendency. If the mean, median, and mode are very similar values, there is a good chance that the data follows a bell-shaped distribution (SPSS command here).

It is also advisable to use a frequency graph too, so you can check the visual shape of your data (If your chart is a histogram, you can add a distribution curve using SPSS: From the menus, choose: Elements > Show Distribution Curve).

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Normal distributions become more apparent (i.e., perfect) the finer the level of measurement and the larger the sample from a population.

You can also calculate coefficients which tell us about the size of the distribution tails in relation to the bump in the middle of the bell curve. For example, Kolmogorov Smirnov and Shapiro-Wilk tests can be calculated using SPSS.

These tests compare your data to a normal distribution and provide a p-value, which, if significant (p < .05), indicates your data is different from a normal distribution (thus, on this occasion, we do not want a significant result and need a p-valuehigher than 0.05).

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Bell Shaped Curve: Normal Distribution In Statistics (2024)

FAQs

Is the curve bell-shaped in normal distribution? ›

Bell curves are visual representations of normal distribution, also called Gaussian distribution. A normal distribution curve, when graphed out, typically follows a bell-shaped curve, hence the name.

How do you know if a bell curve is normally distributed? ›

A normal distribution is determined by two parameters the mean and the variance. A normal distribution with a mean of 0 and a standard deviation of 1 is called a standard normal distribution.

What does it mean when a distribution is bell-shaped? ›

A curve that resembles a cross-section of a bell; therefore, it is symmetrical. Such curves typically represent continuous frequency distributions. A bell curve is a common type of distribution for a variable, also known as the normal distribution.

How does a bell curve relate to statistical significance? ›

The second building block of statistical significance is the normal distribution, also called the Gaussian or bell curve. The normal distribution is used to represent how data from a process is distributed and is defined by the mean, given the Greek letter μ (mu), and the standard deviation, given the letter σ (sigma).

Is a normal distribution bell-shaped True or false? ›

So, we have: Solution: The answer is TRUE. The graph of a normal distribution is symmetric to its mean and is characterized by a bell shape.

What shape is a curve normal distribution? ›

A normal distribution is a true symmetric distribution of observed values. When a histogram is constructed on values that are normally distributed, the shape of columns form a symmetrical bell shape. This is why this distribution is also known as a 'normal curve' or 'bell curve'.

How can I tell if my data is normally distributed? ›

You can test the hypothesis that your data were sampled from a Normal (Gaussian) distribution visually (with QQ-plots and histograms) or statistically (with tests such as D'Agostino-Pearson and Kolmogorov-Smirnov).

What is normal distribution bell curve examples? ›

A good example of a bell curve or normal distribution is the roll of two dice. The distribution is centered around the number seven and the probability decreases as you move away from the center. Here is the percent chance of the various outcomes when you roll two dice.

Why does it matter if data is normally distributed? ›

It is the most important probability distribution in statistics because it accurately describes the distribution of values for many natural phenomena. Characteristics that are the sum of many independent processes frequently follow normal distributions.

Why is the bell-shaped distribution important? ›

Bell-shaped or symmetric, this graph describes the relationship between variables in a data set and the average of the data. Knowing more about this distribution can help you understand how an organisation may analyse large amounts of data and interpret it to make strategic decisions.

What is a real life example of a normal distribution? ›

What are some real life examples of normal distributions? In a normal distribution, half the data will be above the mean and half will be below the mean. Examples of normal distributions include standardized test scores, people's heights, IQ scores, incomes, and shoe size.

What does a bell-shaped distribution say about traits? ›

Bell-shaped distributions tend to result from the summation of many independent influences of small effect, just as traits represent the combination of numerous small genetic and environmental influences.

What is the weakness of normal distribution? ›

The most important problem for the Normal distribution is its name. The name carries the impression that it is the sought to distribution to model any continuous random variable although empirical evidences clearly show that departure from Normality very common than it commonly expected.

How do you interpret a bell curve in statistics? ›

The highest point on the curve, or the top of the bell, represents the most probable event in a series of data (its mean, mode, and median in this case), while all other possible occurrences are symmetrically distributed around the mean, creating a downward-sloping curve on each side of the peak.

What is a good p value for normal distribution? ›

Prism also uses the traditional 0.05 cut-off to answer the question whether the data passed the normality test. If the P value is greater than 0.05, the answer is Yes. If the P value is less than or equal to 0.05, the answer is No.

Is the graph of the standard normal distribution bell-shaped? ›

Symmetricity : — The normal distribution is symmetric about its mean, which means that the probability of observing a value above the mean is the same as the probability of observing a value below the mean. The bell-shaped curve of the normal distribution reflects this symmetry.

Is an inverted bell curve a normal distribution? ›

The reverse bell curve is yet another name for the inverse normal distribution and works in the same manner as the inverse Gaussian distribution. Using the same techniques, it's possible to determine the probability of data taking place between two bounds, rather than just below one bound.

What type of function is a bell curve? ›

Many common probability distribution functions are bell curves. Some bell shaped functions, such as the Gaussian function and the probability distribution of the Cauchy distribution, can be used to construct sequences of functions with decreasing variance that approach the Dirac delta distribution.

What is the symbol for the normal distribution? ›

The Standard Normal Distribution

A random variable that has a standard normal distribution is usually denoted with Z . That is Z∼N(0,1) Z ∼ N ( 0 , 1 ) .

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