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Statistics and probability

Unit 1: analyzing categorical data, unit 2: displaying and comparing quantitative data, unit 3: summarizing quantitative data, unit 4: modeling data distributions, unit 5: exploring bivariate numerical data, unit 6: study design, unit 7: probability, unit 8: counting, permutations, and combinations, unit 9: random variables, unit 10: sampling distributions, unit 11: confidence intervals, unit 12: significance tests (hypothesis testing), unit 13: two-sample inference for the difference between groups, unit 14: inference for categorical data (chi-square tests), unit 15: advanced regression (inference and transforming), unit 16: analysis of variance (anova).

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Compute values for common data set attributes (mean, median, mode, standard deviation, variance, range, interquartile range, skewness, etc.). For help in using the calculator, read the Frequently-Asked Questions or review the Sample Problem .

  • In the unshaded text box, enter a value for each element in the data set.
  • Leave a space between each entry.
  • Click Calculate to compute values for data set attributes.

Frequently-Asked Questions

Instructions: To find the answer to a frequently-asked question, simply click on the question.

What a mean?

A mean score is an average score, often denoted by X . It is the sum of all the scores in a data set divided by the number of observations in the data set. Thus, if you have a set of N numbers ( X 1 , X 1 , X 1 , . . . X N ), the mean of those numbers would be:

X = ( X 1 + X 2 + X 3 + . . . + X N ) / N = [ Σ X i ] / N

For example, the mean of the numbers 1, 2, and 3 would be (1 + 2 + 3)/3 or 2.

What is a median?

The median is a simple measure of central tendency. To find the median, we arrange the observations in order from smallest to largest value. If there is an odd number of observations, the median is the middle value. If there is an even number of observations, the median is the average of the two middle values.

Thus, in a sample of four families, we might want to compute the median annual income. Suppose the incomes are $30,000 for the first family; $50,000, for the second; $90,000, for the third; and $110,000, for the fourth. The two middle values are $50,000 and $90,000. Therefore, the median annual income is ($50,000 + $90,000)/2 or $70,000.

What is a mode?

The mode is the most frequently appearing value in a population or sample.

Suppose we draw a sample of five boys and measure their weights. They weigh 100 pounds, 100 pounds, 130 pounds, 140 pounds, and 150 pounds. Since more boys weigh 100 pounds than any other weight, the sample mode would equal 100 pounds.

What is the standard deviation?

The standard deviation is a numerical value used to indicate how widely individuals in a group vary. If individual observations vary greatly from the group mean, the standard deviation is big; and vice versa.

It is important to distinguish between the standard deviation of a population and the standard deviation of a sample . They have different notation, and they are computed differently. The standard deviation of a population is denoted by σ and the standard deviation of a sample, by s .

The standard deviation of a population is defined by the following formula:

σ = sqrt [ Σ ( X i - μ) 2 / N ]

where σ is the population standard deviation, μ is the population mean, X i is the i th element from the population, and N is the number of elements in the population.

The standard deviation of a sample is defined by slightly different formula:

s = sqrt [ Σ ( x i - x ) 2 / (n - 1) ]

where s is the sample standard deviation, x is the sample mean, x i is the i th element from the sample, and n is the number of elements in the sample.

Note: The standard deviation is the square root of the variance.

What is the variance?

The variance is a numerical value used to indicate how widely individuals in a group vary. If individual observations vary greatly from the group mean, the standard deviation is big; and vice versa.

It is important to distinguish between the variance of a population and the variance of a sample . They have different notation, and they are computed differently. The variance of a population is denoted by σ 2 and the variance of a sample, by s 2 .

The variance of a population is defined by the following formula:

σ 2 = Σ ( X i - μ) 2 / N

where σ 2 is the population variance, μ is the population mean, X i is the i th element from the population, and N is the number of elements in the population.

The variance of a sample is defined by slightly different formula:

s 2 = Σ ( x i - x ) 2 / (n - 1)

where s 2 is the sample variance, x is the sample mean, x i is the i th element from the sample, and n is the number of elements in the sample.

Note: The square root of the variance is the standard deviation.

What is the range?

The range is a simple measure of variation in a data set. It is the difference between the biggest and smallest observation in the data set.

Range = Maximum value - Minimum value

For example, suppose a data set consisted of the following five observations: (3, 5, 5, 7, 8}. The range in that data set would be 8 - 3 or 5.

What is the interquartile range?

The interquartile range (IQR) is a measure of variability, based on dividing a data set into quartiles .

Quartiles divide a rank-ordered data set into four equal parts. The values that divide each part are called the first, second, and third quartiles; and they are denoted by Q1, Q2, and Q3, respectively.

  • Q2 is the median of the data set.
  • Q1 is the "middle" value in the first half of the rank-ordered data set (i.e., the values that come before the median).
  • Q3 is the "middle" value in the second half of the rank-ordered data set.

The interquartile range is equal to Q3 minus Q1.

For example, consider a data set made up of the following nine numbers: 1, 3, 4, 4, 5, 6, 6, 7, 11.

  • Q2 is the median of the rank-ordered data set (i.e., the middle value). In this example, the middle value is 5.
  • Q1 is the middle value in the first half of the data set. Since there are an even number of data points in the first half of the data set, the middle value is the average of the two middle values; that is, Q1 = (3 + 4)/2 or Q1 = 3.5.
  • Q3 is the middle value in the second half of the data set. Again, since the second half of the data set has an even number of observations, the middle value is the average of the two middle values; that is, Q3 = (6 + 7)/2 or Q3 = 6.5.
  • The interquartile range is Q3 minus Q1, so IQR = 6.5 - 3.5 = 3.

What is the lower quartile (Q1)?

What is the upper quartile (q3), what is skewness.

Data sets with fewer observations on the right (toward higher values) are said to be skewed right; and data sets with fewer observations on the left (toward lower values) are said to be skewed left.

Skewness is a quantitative measure of the degree to which a data set skews left or skews right. Here is a formula for computing skewness of sample data:

where Skew s is sample skewness, x i is value for observation i , x is the sample mean, s is the sample standard deviation, and n is the number of observations in the data set.

Here is a formula for computing skewness of population data:

where Skew p is population skewness, x i is value for observation i , μ is the population mean, σ is the population standard deviation, and N is the number of observations in the population.

Data sets that are roughly symmetric have a skewness value near zero. Data sets that skew left have a negative skewness; data sets that skew right have a positive skewness.

Note: There are different ways to measure skewness. This calculator uses the same formula as SPSS and Excel. But, if you use different software, you may get a different result.

Why does the calculator sometimes display "NaN" as an output?

Certain computations require a minimum number of observations. When the number of observations in a data set is insufficient, the calculator displays "NaN" as an output.

Here are the requirements for outputs that require a minimum number of observations:

  • The sample variance and the sample standard deviation each require at least two observations.
  • Population skewness requires at least two observations.
  • Sample skewness requires at least three observations.
  • The lower quartile (Q1), the upper quartile (Q3), and the interquartile range require at least three observations.

Sample Problem

At the end of every school year, the state administers a reading test to a simple random sample of 36 third graders. The test score from each sampled student is shown below:

50, 55, 60, 62, 62, 65, 67, 67, 70, 70, 70, 70, 72, 72, 73, 73, 75, 75, 75, 78, 78, 78, 78, 80, 80, 80, 82, 82, 85, 85, 85, 88, 88, 90, 90, 90 

Find the mean, median, standard deviation, interquartile range, and skewness for the data set.

To solve this problem, enter test scores from the sample data set into the Statistics Calculator. Remember to leave a space (not a comma) between each entry, as shown below:

  • Click the Calculate button to generate the output shown below.
  • From the calculator output, we see that the mean of the data set is 75, the median is 75, the sample standard deviation is about 9.95, the interquartile range is 12, and the sample skewness is about -0.47.

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Word Problem: Rachel has 17 apples. She gives some to Sarah. Sarah now has 8 apples. How many apples did Rachel give her?

Simplified Equation: 17 - x = 8

Word Problem: Rhonda has 12 marbles more than Douglas. Douglas has 6 marbles more than Bertha. Rhonda has twice as many marbles as Bertha has. How many marbles does Douglas have?

Variables: Rhonda's marbles is represented by (r), Douglas' marbles is represented by (d) and Bertha's marbles is represented by (b)

Simplified Equation: {r = d + 12, d = b + 6, r = 2 × b}

Word Problem: if there are 40 cookies all together and Angela takes 10 and Brett takes 5 how many are left?

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  • Symbolab is the best calculator for solving a wide range of word problems, including age problems, distance problems, cost problems, investments problems, number problems, and percent problems.
  • What is an age problem?
  • An age problem is a type of word problem in math that involves calculating the age of one or more people at a specific point in time. These problems often use phrases such as 'x years ago,' 'in y years,' or 'y years later,' which indicate that the problem is related to time and age.

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  • Middle School Math Solutions – Equation Calculator Welcome to our new "Getting Started" math solutions series. Over the next few weeks, we'll be showing how Symbolab... Read More

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  1. How to Solve Statistics Problems Accurately

    solve statistics problems online free

  2. How to Solve Statistics Problems in Real Life Like A Pro

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  3. 😝 Solve statistics problems online. Solve statistics problems online

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  4. ⚡ Statistics math solver. Statistics Problem Solver Online. 2022-10-13

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  5. Statistics Practice Problems Chapter 1 Answers

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  6. How to Solve Probability Problems in Statistics

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VIDEO

  1. Mathematical Statistics, Lecture 0

  2. What is Statistics?

  3. 3.1 Basic Statistics

  4. Statistics and Numerical methods

  5. formulas of statistics

  6. Learn BASICS of STATISTICS

COMMENTS

  1. Mathway

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  2. Statistics Calculator

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    Symbolab is the best step by step calculator for a wide range of math problems, from basic arithmetic to advanced calculus and linear algebra. It shows you the solution, graph, detailed steps and explanations for each problem. Is there a step by step calculator for physics?

  5. Statistics Problems

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  6. Statistics and Probability

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  7. Microsoft Math Solver

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  9. Statistics Calculator with Steps

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  10. Statistics Calculator

    Click the Calculate button to generate the output shown below. From the calculator output, we see that the mean of the data set is 75, the median is 75, the sample standard deviation is about 9.95, the interquartile range is 12, and the sample skewness is about -0.47.

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    More Topics ». Made possible by the Wolfram Language—building on 30+ years of research & development ». Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. For math, science, nutrition, history, geography, engineering, mathematics, linguistics, sports, finance, music….

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  14. Free Online Statistics Calculators

    :: Statistics and probability Statistics and probability There are 7 calculators in this category Descriptive Statistics Calculator - Find Arithmetic mean, mode, median, minimum, maximum of a data set. Standard Deviation Calculator - Find standard deviation, variance and range of a data set.

  15. Symbolab Math Calculator

    Symbolab, Making Math Simpler. Word Problems. Provide step-by-step solutions to math word problems. Graphing. Plot and analyze functions and equations with detailed steps. Geometry. Solve geometry problems, proofs, and draw geometric shapes. Math Help Tailored For You.

  16. Statistics Calculator For Stat Problems: (100% Free)

    Solve common statistics problems like probability, mean, mode, median, quartile, hypothesis testing and more with these free online calculators. Choose from a wide range of calculators for researchers, professionals and students to get the solution of your statistical problems.

  17. Free Statistics Calculator

    Problem Solvers Statistics Calculator Free Statistics Calculator Enter data separated by either commas or semicolons. Sorted data: 2, 4, 4, 4, 5, 5, 7, 9 Formulas Given a set of n numbers: \displaystyle \left\ { x_ {1},x_ {2},x_ {3},x_ {4}......,x_ {n}\right\} {x1,x2,x3,x4......,xn} Mean

  18. StatsText.Com

    This website is a free, comprehensive and interactive statistics online textbook. Features of this online statistics text: Free. That's right. No cost. ... If you only want to solve statistics problems step-by-step, then you can use , where you can work step-by-step through more than 77 statistical analyses. You can generate hundreds of ...

  19. MathGPT

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  20. Math Problem Solver

    Problem Solver Subjects. Our math problem solver that lets you input a wide variety of math math problems and it will provide a step by step answer. This math solver excels at math word problems as well as a wide range of math subjects. Here are example math problems within each subject that can be input into the calculator and solved.

  21. Practice Problems to prepare for your exams

    The collection contains solved statistic problems of various different areas in statistics, such as Descriptive Statistics, Confidence Intervals, Calculation of Normal Probabilities, Hypothesis Testing, Correlation and Regression, and Analysis of Variance (For a list of 30,00+ step-by-step solved math problems, click here )

  22. Probability Problems Calculator

    Free Probability Problems Calculator - solve probability word problems step by step

  23. Word Problems Calculator

    Symbolab is the best calculator for solving a wide range of word problems, including age problems, distance problems, cost problems, investments problems, number problems, and percent problems. Show more Why users love our Word Problems Calculator Middle School Math Solutions - Inequalities Calculator