Standard Normal Table

by
Post Views: 4,025

A standard normal table, also called the unit normal table or Z-table, is a table for the values of Φ calculated mathematically, and these are the values from the cumulative normal distribution function. A standard normal distribution table is used to find the probability that a statistic is observed below, above, or between values on the standard normal distribution, and by extension, any normal distribution. Since probability tables cannot be printed for every normal distribution, as there is an infinite variety (families) of normal distributions, it is common practice to convert a normal distribution to a standard normal and then use the standard normal table to find the required probabilities (area under the normal curve).

Standard Normal Curve

The standard normal curve is symmetrical, so the table can be used for values going in any direction, for example, a negative 0.45 or positive 0.45 has an area of 0.1736.

The Standard Normal distribution is used in various hypothesis testing procedures such as tests on single means, the difference between two means, and tests on proportions. The Standard Normal distribution has a mean of 0 and a standard deviation of 1.

The values inside the given table represent the areas under the standard normal curve for values between 0 and the relative z-score.

The table value for $$Z is 1 minus the value of the cumulative normal distribution.

Standard Normal Table (Area Under the Normal Curve)

Standard Normal Table

For example, the value for 1.96 is $P(Z>1.96) = 0.0250$.

Standard Normal Table (Summary)

  • A table of values for the cumulative distribution function (CDF) of the standard normal distribution.
  • The standard normal distribution has a mean of 0 and a standard deviation of 1.
  • This table shows the probability that a standard normal variable will be less than a certain value (z-score).
https://itfeature.com

Real Life Applications of Standard Normal Distribution

The standard Normal Table (also called the Z Table) helps find probabilities for normally distributed data. It helps in decision-making, risk assessment, and quality control. The following are real-life practical applications (education, manufacturing, healthcare, finance, and social sciences):

  1. Exam Scores and Grading: A class has normally distributed exam scores with a mean score of 75 and a standard deviation of 10. One can compute what percentage of students scored (say) above 85?
  2. Quality Control (Manufacturing): In a manufacturing process, bolts are produced with normally distributed lengths having a mean = 5cm and a standard deviation = 0.2cm. If bolts shorter than 4.8 cm are rejected, what percentage is defective?
  3. Medical Health Studies: Blood pressure readings are normally distributed with population mean = 120mmHg and population standard deviation = 15mmHg. What percentage of people have a reading below 140 mmHg?
  4. Stock Returns (Finance & Investment): An investor has annual returns normally distributed with a mean of 8% and a standard deviation of 5%. What is the percentage of returns less than 2%?
  5. Height Distribution (Demographics): Adult male heights are normally distributed with a population mean = 70 inches and a standard deviation = 3 inches. What percentage of men are taller than 77 inches?

FAQs about Standard Normal Table

  1. What is a standard normal distribution table?
  2. What is the value of the mean and variance in a standard normal distribution?
  3. What is the cumulative distribution function of the standard normal distribution?
  4. What kind of values are in the standard normal distribution table?
  5. Is the standard normal distribution curve symmetrical?
  6. What is meant by the area under the normal curve?
  7. What is the use of the standard normal distribution?
  8. The values of $Z$ inside the standard normal table range from 0 to what value?

For further details, see Standard Normal

See about the measure of asymmetry

Probability in R Language

Histogram Graph

by
Post Views: 632

A histogram is very similar to a bar chart for a frequency distribution based on quantitative data, showing the distribution of qualitative data. It is a useful graphical representation of data that helps to visualize the distribution of data.

Important Points to Draw a Histogram Graph

The histogram is constructed from the grouped data by taking the class boundaries (not class limits) along the x-axis and the corresponding frequencies along the y-axis. For ungrouped data, we have to form the grouped frequency distribution before making a histogram. It consists of a set of bars (like a bar chart), but these bars are adjacent to each other, and the height of the bars is proportional to the frequency associated with respective classes.

The area of each rectangle represented the respective class frequencies. When the class intervals are equal, the rectangles all have the same width, and their heights directly represent the class frequencies. For the case in which class intervals are not all equal, the height of the rectangle (bar) over an unequal class interval is to be adjusted because it is the area and not the height that measures frequency. This means that the height of a rectangle must be proportionally decreased if the length of the corresponding class interval increases.

For example, if the length of a class interval becomes double, then the height of the rectangle is to be halved so that the area, being the fundamental property of the rectangle of the histogram, remains unchanged. This sort of rescaling is necessary to observe the correct pattern of distribution.

Important Features of Histogram

The important feature of the Histogram graph is that there is no gap (space) between the vertical bars because the variable plotted on the horizontal axis is quantitative and the variable is from the measure of scale, either interval or ratio. Thus, it provides an easily interpreted visual representation of a frequency distribution. Note that class midpoints are used as labels for the classes.

It allows us to analyze extremely large datasets by reducing them to a single graphical representation, which is used to show primary, secondary, and tertiary peaks in data, and also helps us by giving a visual representation of the statistical significance of those peaks.

Alternative to Histogram

An alternative to the histogram is kernel density estimation, which uses a kernel to smooth samples. This will construct a smooth probability density function, which will, in general, more accurately reflect the underlying variable.

Histograms for Continuous Grouped Data

To draw a histogram graph from the continuous grouped frequency distribution, the following steps are taken.

  1. Mark the class boundaries of the classes along the x-axis.
  2. Mark frequencies along the y-axis.
  3. Draw a rectangle for each class such that the height of each rectangle is proportional to the frequency corresponding to that class. This is the case when classes are of equal width, as they often are.
  4. If the classes are of unequal width, then the area instead of the height of each rectangle is proportional to the frequency corresponding to that class, and the height of each rectangle is obtained by dividing the frequency of the class by the width of that class.

It may be noted that the area under a histogram graph can be calculated by adding up the areas of all the rectangles that constitute the histogram. The area of one rectangle is obtained by the multiplication of the width of the class by the corresponding frequency, i.e.

Area of a single rectangle = width of the class x frequency of the class

Histogram for Discrete Data

Bar graphs are usually drawn for discrete and categorical data, but there are some situations where there is a need to make an approximation; histograms may be constructed. To construct a histogram graph for discrete grouped data, the following steps are taken:

  1. Mark possible values on the x-axis.
  2. Mark frequencies along the y-axis.
  3. Draw a rectangle centered on each value with equal width on each side, possibly 0.5 to either side of the value.
Histogram graph

The advantages of the histograms as compared to the unprocessed data are:

  1. It gives a range of data.
  2. It gives the location of the data.
  3. It gives a clue about the skewness of the data.
  4. It gives information about the out-of-control situation.
  5. Histograms are density estimates (give a good impression of the distribution of data.
  6. It can be compared to the normal curve.

The disadvantages are:

  • Exact values cannot be read from the histogram graph because the data is grouped into categories, and the individuality of the data vanishes in grouped data.
  • It is more difficult to compare two data sets.
  • It is used only for the continuous data set.

FAQs about Histogram

  1. What is a histogram graph?
  2. What is the difference between a bar chart and a histogram?
  3. What are the important features of histograms?
  4. What are the advantages and disadvantages of histogram graphs?
  5. How can one draw a histogram for a discrete data set?
  6. How can one draw a histogram for a continuous data set?

Graphical Representation of Data, Data Visualization/ Graphics in R

Correlation Coefficient Range

by
Post Views: 1,354

The coefficient of correlation (r) measures the strength and direction of a linear relationship between two variables. In this post, we will discuss about coefficient of correlation and the coefficient of determination.

Correlation Coefficient Ranges

The correlation coefficient ranges from -1 to +1, where a value of +1 indicates the perfect positive correlation (as one variable increases, the other increases proportionally), the -1 value indicates the perfect negative correlation (as one variable increases, the other decreases proportionally), and the value of 0 indicates no linear correlation (no relationship between the variables).

The coefficient of correlation values between -1 and +1 indicate the degree of strength and direction of rthe elationship:

The strength of correlation depends on the absolute value of r:

Range of Correlation ValueInterpretation
0.90 to 1.00Very strong correlation
0.70 to 0.89Strong correlation
0.40 to 0.69Moderate correlation
0.10 to 0.39Weak correlation
0.00 to 0.09No or negligible correlation

The closer the value of the correlation coefficient is to ±1, the stronger the linear relationship.

Coefficient of Determination

We know that the ratio of the explained variation to the total variation is called the coefficient of determination, which is the square of the Correlation Coefficient Range and lies between $-1$ and $+1$. This ratio (coefficient of determination) is non-negative; therefore, denoted by $r^2$, thus

\begin{align*}
r^2&=\frac{\text{Explained Variation}}{\text{Total Variation}}\\
&=\frac{\sum (\hat{Y}-\overline{Y})^2}{\sum (Y-\overline{Y})^2}
\end{align*}

It can be seen that if the total variation is all explained, the ratio $r^2$ (Coefficient of Determination) is one, and if the total variation is all unexplained, then the explained variation and the ratio $r^2$ are zero.

The square root of the coefficient of determination is called the correlation coefficient, given by

\begin{align*}
r&=\sqrt{ \frac{\text{Explained Variation}}{\text{Total Variation}} }\\
&=\pm \sqrt{\frac{\sum (\hat{Y}-\overline{Y})^2}{\sum (Y-\overline{Y})^2}}
\end{align*}

and

\[\sum (\hat{Y}-\overline{Y})^2=\sum(Y-\overline{Y})^2-\sum (Y-\hat{Y})^2\]

Therefore

\begin{align*}
r&=\sqrt{ \frac{\sum(Y-\overline{Y})^2-\sum (Y-\hat{Y})^2} {\sum(Y-\overline{Y})^2} }\\
&=\sqrt{1-\frac{\sum (Y-\hat{Y})^2}{\sum(Y-\overline{Y})^2}}\\
&=\sqrt{1-\frac{\text{Unexplained Variation}}{\text{Total Variation}}}=\sqrt{1-\frac{S_{y.x}^2}{s_y^2}}
\end{align*}

where $s_{y.x}^2=\frac{1}{n} \sum (Y-\hat{Y})^2$ and $s_y^2=\frac{1}{n} \sum (Y-\overline{Y})^2$

\begin{align*}
\Rightarrow r^2&=1-\frac{s_{y.x}^2}{s_y^2}\\
\Rightarrow s_{y.x}^2&=s_y^2(1-r^2)
\end{align*}

Since variances are non-negative

\[\frac{s_{y.x}^2}{s_y^2}=1-r^2 \geq 0\]

Solving for inequality, we have

\begin{align*}
1-r^2 & \geq 0\\
\Rightarrow r^2 \leq 1\, \text{or}\, |r| &\leq 1\\
\Rightarrow & -1 \leq r\leq 1
\end{align*}

Therefore, the Correlation Coefficient Range lies between $-1$ and $+1$ inclusive.

Correlation Coefficient Range

Alternative Proof: Correlation Coefficient Range

Since $\rho(X,Y)=\rho(X^*,Y^*)$ where $X^*=\frac{X-\mu_X}{\sigma_X}$ and $Y^*=\frac{Y-Y^*}{\sigma_Y}$

and as covariance is bi-linear and $X^*, Y^*$ have zero mean and variance 1, therefore

\begin{align*}
\rho(X^*,Y^*)&=Cov(X^*,Y^*)=Cov\{\frac{X-\mu_X}{\sigma_X},\frac{Y-\mu_Y}{\sigma_Y}\}\\
&=\frac{Cov(X-\mu_X,Y-\mu_Y)}{\sigma_X\sigma_Y}\\
&=\frac{Cov(X,Y)}{\sigma_X \sigma_Y}=\rho(X,Y)
\end{align*}

We also know that the variance of any random variable is $\ge 0$; it could be zero, i.e., $(Var(X)=0)$ if and only if $X$ is a constant (almost surely), therefore

\[V(X^* \pm Y^*)=V(X^*)+V(Y^*)\pm2Cov(X^*,Y^*)\]

As $Var(X^*)=1$ and $Var(Y^*)=1$, the above equation would be negative if $Cov(X^*,Y^*)$ is either greater than 1 or less than -1. Hence \[1\geq \rho(X,Y)=\rho(X^*,Y^*)\geq -1\].

If $\rho(X,Y)=Cov(X^*,Y^*)=1$ then $Var(X^*- Y^*)=0$ making $X^*= Y^*$ almost surely. Similarly, if $\rho(X,Y )=Cov(X^*,Y^*)=-1$ then $X^* = – Y^*$ almost surely. In either case, $Y$ would be a linear function of $X$ almost surely.

For proof of Cauchy-Schwarz Inequality, please follow the link

We can see that the Correlation Coefficient range lies between $-1$ and $+1$.

Coefficient of Correlation Range

Real-Life Example

Variable 1Variable 2Coefficient ValueInterpretation
Study hoursExam scores+0.85Strong positive
Screen timeSleep duration-0.70Strong negative
AgeShoe size~0.00No linear correlation

FAQs about Correlation Coefficient

  • What is a coefficient of correlation?
  • What does a positive or negative correlation mean?
  • What is a strong or weak correlation?
  • Can correlation imply causation?
  • What are the types of correlation coefficients?
  • When should I use Pearson vs. Spearman correlation?
  • What are the assumptions of the Pearson correlation?
  • Can correlation be used for more than two variables?
  • How is correlation different from regression?
  • How is the correlation coefficient calculated?
  • What does a zero correlation mean?
  • Can correlation be misleading?

Learn more about