## Normal Distribution And Histogram In R

I spent much time lately seeking for a tool that would allow me to easily draw a histogram with a normal distribution curve on the same diagram. I could create the histogram in OOCalc, by using the FREQUENCY function and creating a column chart, but I found no way to add a curve, so I gave up. I started searching for something more powerful than OpenOffice. Of course, no Windows applications were allowed.

I googled my problem up before trying to use Maxima or something similar, and I found R.I havent heard about the R project earlier, but I decided to give it a try. And it was worth trying. Even if I were able to do the same in Octave or Maxima, I dont think it could have been done easier.

## How To Create A Normal Distribution Curve Within Tableau

This past week during the course of my normal day-to-day activities at work, I came across a requirement for creating a normal distribution curve within Tableau. I immediately thought of **Jim Dehners **blog post on How-To Create a Normal Distribution Chart and decided to replicate it. This blog post is about my understanding of his original work and the steps that I followed to replicate it.I do not take credit for the original idea and forever thankful for his incredible contribution to the tableau community

Tableau Software provides users with different types of analysis so they can visualize more easily their recorded values. One of the most common analyses used to observe the deviations that occur is the Histogram with normal distribution. This type of analysis is represented by the presence of a curve, also called the bell curve. **The normal distribution is also known as the Gaussian distribution and represents the distribution of symmetric probabilities compared to the mean.****Thus, data that are closest to the higher point of the curve have a higher frequency of occurrence than those that are further away.**

## Calculating The Probability Of Normal Distribution

Given the mean of 3 and the standard deviation of 2, we can find the probability of .

In norm.cdf, the location keyword specifies the mean and the scale keyword specifies the standard deviation.

from scipy.stats import normlessthan2=norm.cdfprint

Lets plot a graph.

fig, ax = plt.subplots# for distribution curvex= np.arangeax.fill_between,alpha=0.5, color='g')# for textax.text, fontsize=20)plt.show

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## Probability Questions Using The Standard Model

Questions about standard normal distribution probability can *look* alarming but the key to solving them is understanding what the area under a standard normal curve represents. The total area under a standard normal distribution curve is 100% . For example, the left half of the curve is 50%, or .5. So the probability of a random variable appearing in the left half of the curve is .5.

Of course, not all problems are quite *that* simple, which is why theres a z-table. All a z-table does is measure those probabilities and put them in standard deviations from the mean. The mean is in the center of the standard normal distribution, and a probability of 50% equals zero standard deviations.

## Normal Distribution And Specifications

Now suppose a customer has decided that the upper specification limit for your process should be 112. You can easily see from the histogram and the normal distribution that some of your product will be out of specification. The question is how much will be out of specification. This is where the z value becomes important. z is defined by the equation:

For our example, x = 112, so

z = /10 = 12/10 = 1.2

z represents the number of standard deviations some value is away from the average. So, 112 is 1.2 standard deviations above the average. If z is negative, it means that the value is below the average.

To find out how much product is more than 1.2 standard deviations above the average, you can use what is called the “z table.” The z table gives the fraction of process output that is beyond some value x that is z standard deviations from the average. The z table is given below.

**z Table: Standard Normal Distribution**

The table above returns a value of .1151 for a value of z = 1.2. This means that 11.51% of the data will be above the upper specification limit of 112. This is shown in the figure below. The area in red represents material that is out of specifications on the high side.

You can also use the NORMSDIST function in Excel to find the above result. This function in Excel returns the fraction of results less than a value. So, you must use 1 – NORMSDIST if you want the fraction of data above z. In this example: 1- NORMSDIST = .11507

z = /10 = -17/10 = -1.7

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## Why Do Normal Distributions Matter

All kinds of variables in natural and social sciences are normally or approximately normally distributed. Height, birth weight, reading ability, job satisfaction, or SAT scores are just a few examples of such variables.

Because normally distributed variables are so common, many statistical tests are designed for normally distributed populations.

Understanding the properties of normal distributions means you can use inferential statistics to compare different groups and make estimates about populations using samples.

## Normal Distribution Word Problems Less Than: Steps

**Step 1:***Break up the word problem into parts*:

Plus, you will have EITHER:

**Step 2:** *Draw a picture* to help you visualize the problem. The following graph shows a mean of 15, and an area under 4):

**Step 3:** *Find the z value* by plugging the given values into the formula. The X in our sample graph is 4, and the is 15. You can get these figures from your answers in step 1, where you identified the parts of the problem:

All you have to do to solve the formula is:

**Step 4:***Take the number from step 3, then use the **z-table* to find the area.

**Step 5:***To find a probability, go to step 6a. To find a number from a specific given sample size, go to step 6b.*

**Step 6a**

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## Example: In That Same School One Of Your Friends Is 185m Tall

You can see on the bell curve that 1.85m is **3 standard deviations** from the mean of 1.4, so:

Your friend’s height has a “z-score” of 3.0

It is also possible to **calculate** how many standard deviations 1.85 is from the mean

*How far is 1.85 from the mean?*

It is 1.85 – 1.4 =** 0.45m from the mean**

*How many standard deviations is that?* The standard deviation is 0.15m, so:

0.45m / 0.15m = **3 standard deviations**

So to convert a value to a Standard Score :

- first subtract the mean,
- then divide by the Standard Deviation

And doing that is called “Standardizing”:

We can take any Normal Distribution and convert it to The Standard Normal Distribution.

## Evaluating Set Of Normally Distributed Numbers

When you have a set of numbers, R provides functions for evaluatingthe distribution of those numbers.

Two fundamental functions are **mean** and **sd**,which return the mean and standard deviation of a sequence of numbers, respectively.This example is a final point tally for a geography class:

> points = c> mean 460.3793> sd 81.33239> hist

As is obvious from the histogram, the distribution is not a smooth bell.The normal curve is theoretical and complex reality will rarely line upperfectly with that theoretical construct. In this example, there were a fewhigh-performing students, and a number of students who struggled, stretchingthe **left tail** of the bell curve out a bit.

The shifting of the mean of a bell curve off-center is called **skew**.The **skewness** function in the **e1071** library will evaluatehow skewed a curve is. In this example, the positive value for skew fitsour perception that the center of the curve is shifted to the right.

> library> skewness 0.1320943

If you do not have the e1071 library installed on your system, youcan install it with:

> install.packages

Another way in which a distribution can be distorted away fromnormal is **kurtosis**, which indicates whether the middle isstretched up higher or lower than would be in a clean normal distribution.The negative value returned from the **kurtosis** function for the distribution above indicates that the distribution is more spread out than it would be if it were perfectly normal.

> kurtosis -0.4461787

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## What Is A Bell Curve

A bell curve is a common type of distribution for a variable, also known as the normal distribution. The term “bell curve” originates from the fact that the graph used to depict a normal distribution consists of a symmetrical bell-shaped curve.

The highest point on the curve, or the top of the bell, represents the most probable event in a series of data , while all other possible occurrences are symmetrically distributed around the mean, creating a downward-sloping curve on each side of the peak. The width of the bell curve is described by its standard deviation.

## How To Use The Normal Distribution

Suppose you are interested in a certain quality characteristic, X. You have been monitoring this characteristic using an X-R chart. Both the X chart and the range chart are in statistical control. This means that you can predict what your process will make in the near future. You also know that you have good estimates of the process average and the process standard deviation . Suppose that X = 100 and that = 10.

In addition, you have constructed a histogram for the last month’s data. The histogram is shown below.

This histogram appears to be bell-shaped so you assume that you are dealing with a normal distribution. You can then draw the normal distribution for this process because you know the average and standard deviation . The normal distribution for this process is shown below.

A normal distribution has the following properties:

- 68% of the data is within +/- 1 standard deviation of the average
- 95% of the data is within +/- 2 standard deviations of the average
- 99.7% of the data is within +/- 3 standard deviations of the average

For your process, the following calculations can be done:

Thus, 68% of the data lies between 90 and 110 95% of the data between 80 and 120 and 99.7% of the data between 70 and 130. The specifications for the process have been 65 to 140. Life is good – everything has been within specifications.

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## Kernel Density Estimation Pitfalls

KDE plots have many advantages. Important features of the data are easy to discern , and they afford easy comparisons between subsets. But there are also situations where KDE poorly represents the underlying data. This is because the logic of KDE assumes that the underlying distribution is smooth and unbounded. One way this assumption can fail is when a varible reflects a quantity that is naturally bounded. If there are observations lying close to the bound , the KDE curve may extend to unrealistic values:

sns.displot

This can be partially avoided with the cut parameter, which specifies how far the curve should extend beyond the extreme datapoints. But this influences only where the curve is drawn the density estimate will still smooth over the range where no data can exist, causing it to be artifically low at the extremes of the distribution:

sns.displot

The KDE approach also fails for discrete data or when data are naturally continuous but specific values are over-represented. The important thing to keep in mind is that the KDE will *always show you a smooth curve*, even when the data themselves are not smooth. For example, consider this distribution of diamond weights:

diamonds=sns.load_datasetsns.displot

While the KDE suggests that there are peaks around specific values, the histogram reveals a much more jagged distribution:

sns.displot

sns.displot

## Set Up The Label Table

Technically, you have your bell curve. But it would be hard to read as it lacks any data describing it.

Lets make the normal distribution more informative by adding the labels illustrating all the standard deviation values below and above the mean .

For that, set up yet another helper table as follows:

First, copy the Mean value next to the corresponding cell in column X-Value .

Next, compute the standard deviation values below the mean by entering this simple formula into **cell I4**:

1 | =I5-$F$2 |

Simply put, the formula subtracts the sum of the preceding standard deviation values from the mean. Now, drag the fill handle upward to copy the formula into the remaining two cells .

Repeat the same process for the standard deviations above the mean using the mirror formula:

1 | =I5+$F$2 |

In the same way, execute the formula for the other two cells .

Finally, fill the y-axis label values with zeros as you want the data markers placed on the horizontal axis.

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## Example : Normal Distribution With Mean = 0 And Standard Deviation = 1

To create a normal distribution plot with mean = 0 and standard deviation = 1, we can use the following code:

#Create a sequence of 100 equally spaced numbers between -4 and 4x < - seq#create a vector of values that shows the height of the probability distribution#for each value in xy < - dnorm#plot x and y as a scatterplot with connected lines and add#an x-axis with custom labelsplotaxis)

This generates the following plot:

## Find The Standard Deviation

One down, one to go. Fortunately, Excel has a special function to do all the dirty work of finding the standard deviation for you:

1 | =STDEV.P |

Again, the formula picks all the values from the specified cell range and computes its standard deviationjust dont forget to round up the output as well.

1 | =ROUND,0) |

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## Histogram Using Scatter Chart

Overlaying a normal curve is a little trickier, firstly, the above column chart cant be used and the histogram must be produced using a scatter chart.

Select the data and produce a scatter chart with smooth lines.

Select the chart and click on the ribbon menu, Layout, then Error Bars and then More Error Bars Options. Select Display Direction Minus, End Style No Cap and Error Amount Percentage 100%.

This will produce a scatter chart with the following error bars.

Increase the Line Style Width so that it starts looking like a histogram with no gaps.

Finally, remove the scatter chart line.

## Standard Normal Distribution: How To Find Probability

**Step 1:****Draw a bell curve** and shade in the area that is asked for in the question. The example below shows z > -0.8. That means you are looking for the probability that z is greater than -0.8, so you need to draw a vertical line at -0.8 standard deviations from the mean and shade everything thats greater than that number.

shaded area is z > -0.8

**Step 2:****Visit the normal probability area index **and find a picture that looks like your graph. Follow the instructions on that page to find the z-value for the graph. The z-value *is *the probability.

**Tip: **Step 1 is technically optional, but its *always* a good idea to sketch a graph when youre trying to answer probability word problems. Thats because most mistakes happen not because you cant do the math or read a z-table, but because you subtract a z-score instead of adding (i.e. you imagine the probability under the curve in the wrong direction. A sketch helps you cement in your head exactly what you are looking for.

If youre still having trouble, check out the tutors at Chegg.com. Your first 30 minutes with a live tutor is free!

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## Example : Normal Distribution With Customized Mean And Standard Deviation

To create a normal distribution plot with a user-defined mean and standard deviation, we can use the following code:

#define population mean and standard deviationpopulation_mean < - 50population_sd < - 5#define upper and lower boundlower_bound < - population_mean - population_sdupper_bound < - population_mean + population_sd#Create a sequence of 1000 x values based on population mean and standard deviationx < - seq * population_sd + population_mean#create a vector of values that shows the height of the probability distribution#for each value in xy < - dnorm#plot normal distribution with customized x-axis labelsplotsd_axis_bounds = 5axis_bounds < - seqaxis

This generates the following plot:

## How To Plot A Normal Distribution In Python

To plot a normal distribution in Python, you can use the following syntax:

#x-axis ranges from -3 and 3 with .001 stepsx = np.arange#plot normal distribution with mean 0 and standard deviation 1plt.plot)

The **x** array defines the range for the x-axis and the **plt.plot** produces the curve for the normal distribution with the specified mean and standard deviation.

The following examples show how to use these functions in practice.

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## Begin By Drawing A Horizontal Line Axis

Sketch a picture of a normal distribution. Sketch a picture of a normal distribution. This video will show you how to draw the normal distribution and the standard normal. Begin by drawing a horizontal line axis. This line represents the overall average. When drawing the normal distribution.

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Then draw a vertical line from the horizontal axis through the center of the curve cutting it in half. This line represents the overall average. This video will show you how to draw the normal distribution and the standard normal. Next draw a normal bell-shaped curve centered on the horizontal axis. Begin by drawing a horizontal line axis. Normal Distribution Bpi Consulting.

Next draw a normal bell-shaped curve centered on the horizontal axis. When drawing the normal distribution. Then draw a vertical line from the horizontal axis through the center of the curve cutting it in half. Sketch a picture of a normal distribution. Next draw a normal bell-shaped curve centered on the horizontal axis. Normal Distributions Review Article Khan Academy.

Sketch a picture of a normal distribution. This line represents the overall average. Sketch a picture of a normal distribution. Next draw a normal bell-shaped curve centered on the horizontal axis. When drawing the normal distribution. The Standard Normal Distribution Introduction To Statistics.