Important formulas and processes to understand and memorize

Question 1

What are the three Measures of Variation

Answer 1

Range, Variance, and Standard Diviation

Question 2

How do you obtain the range of a population or a sample of measurements?

Answer 2

The largest measurement minus the smallest measurement

Question 3

What is a Diviation

Answer 3

In mathematics and statistics, deviation is a measure of difference between the observed value of a variable and some other value, often that variable’s mean.

Question 4

How do you calculate Population Variance

Answer 4

Question 5

What is the population standard diviation?

Answer 5

Question 6

Population Variance formula

Answer 6

Question 7

Sample Variance and Sample Standard Diviation

Answer 7

Question 8

What are the commands to calculate the variance of a population/sample

Answer 8

Question 9

Steps in calculating the standard deviation

Answer 9

Step 1: Find the mean.
Step 2: For each data point, find the square of its distance to the mean.
Step 3: Sum the values from Step 2.
Step 4: Divide by the number of data points.
Step 5: Take the square root.

Question 10

What are the excel functions for standard deviation and sample standard deviation?

Answer 10

Question 11

What is a tolerance Interval?

Answer 11

An interval of numbers that contains a specified percentage of the individual measurements in a population.

Question 12

The Emperical Rule for a normally distributed population

Answer 12

Question 13

What is a z-score and how is it calculated?

Answer 13

Question 14

How to calculate z-score on excel

Answer 14

First, we need to calculate the Mean (illustrated in cell D2 in Figure 1 below) and the Standard Deviation (illustrated in cell E2 below) because we’ll need the Mean and Standard Deviation to calculate the z-score of each value in our data set.

Then, to calculate each z-score, we first create a new Column Header called Z-Scores, as illustrated in Column B of Figure 1 below. Next we build a formula.

Notice the dollar signs $ in the formula. These dollar signs are what are known as absolute markers. They tell the formula to always look in a specific cell for a value, even if the formula is copied into another cell.

Question 15

How do you make a pie chart on excel?

Use this question as an example: A multiple choice question on an exam has four possible responses—(a), (b), (c), and (d).

When 260 students take the exam, 26 give response (a), 26 give response (b), 104 give response (c), and 104 give response (d). Consider constructing a pie chart for the exam question responses.”

Answer 15

Enter Data:

In an Excel spreadsheet, create two columns. In the first column, label the responses (a), (b), (c), and (d).
In the second column, enter the corresponding counts: 26 for (a), 26 for (b), 104 for (c), and 104 for (d).
Your data should look like this:

Create the Pie Chart:

Select the data in both columns (including the labels and counts).
Go to the “Insert” tab in Excel.
In the “Charts” group, click on “Pie” and select the type of pie chart you want to create (e.g., 2-D Pie or 3-D Pie).
Customize the Pie Chart (optional):

Double-click on the chart elements (slices, labels, title, etc.) to make any necessary adjustments to the chart’s appearance or labels.

You can add a title, change the colors, or explode a slice (pull it out) to emphasize a particular response if needed.

Finalize and Save:

Once you are satisfied with the appearance of your pie chart, you can save or export it as needed.

A | B |
|——-|——-|
| (a) | 26 |
| (b) | 26 |
| (c) | 104 |
| (d) | 104 |

Question 16

How would you calculate the proportions, angles and degrees on a pie chart to determine percentages of a population

Example question:

A multiple choice question on an exam has four possible responses—(a), (b), (c), and (d). When 260 students take the exam, 26 give response (a), 26 give response (b), 104 give response (c), and 104 give response (d). Consider constructing a pie chart for the exam question responses.

(a) How many degrees (out of 360) would be assigned to the “pie slice” for response (a)?

Answer 16

Calculate the Proportions:

Calculate the proportion of students who chose each response by dividing the count for each response by the total number of students (260).

For example:

Proportion of (a) responses = 26 / 260 = 0.1
Proportion of (b) responses = 26 / 260 = 0.1
Proportion of (c) responses = 104 / 260 = 0.4
Proportion of (d) responses = 104 / 260 = 0.4
Convert Proportions to Degrees:

To convert the proportions to degrees, use the fact that a full circle has 360 degrees.

Multiply each proportion by 360 to get the corresponding angle in degrees.

For example:

Angle for (a) responses = 0.1 * 360 = 36 degrees
Angle for (b) responses = 0.1 * 360 = 36 degrees
Angle for (c) responses = 0.4 * 360 = 144 degrees
Angle for (d) responses = 0.4 * 360 = 144 degrees
Draw the Pie Chart:

Draw a circle to represent the whole pie.
Divide the circle into four sections (one for each response), and label each section with the corresponding response (a, b, c, d).
Use a protractor or a compass to measure and draw each section’s angle according to the degrees you calculated in step 2.
Label and Customize (optional):

You can label each section with the response letter and its corresponding angle.
If you want to add color or shading for emphasis, you can do so manually.
Finalize:

Review your pie chart to ensure that the angles accurately represent the distribution of student responses.

Question 17

In what kind of senarious would I want to use range, variation, and standard deviation to solve?

Answer 17

Range, variation, and standard deviation are statistical measures that provide insights into the dispersion, spread, and variability of data. You might use these measures in various scenarios to understand and analyze data in different contexts, including:

Quality Control in Manufacturing:

Range, variation, and standard deviation can help monitor and control the consistency and quality of manufactured products. For example, in the production of electronic components, you might use these measures to assess the variability in product specifications or defects.
Financial Analysis:

In finance, you can use these measures to analyze the volatility and risk associated with investment portfolios. Standard deviation, in particular, is commonly used to assess the variability of returns on financial assets.
Healthcare and Medical Research:

Variation and standard deviation can be used to analyze patient data and assess the spread of health-related measurements, such as blood pressure, cholesterol levels, or drug effectiveness. In clinical trials, standard deviation can indicate how consistent the results are among participants.
Market Research:

When conducting surveys or market research, you may use these measures to understand the variability in responses or consumer preferences. For example, if you’re analyzing customer satisfaction scores, standard deviation can indicate how opinions vary among respondents.
Education and Assessment:

In educational settings, you might use variation and standard deviation to evaluate the performance of students on tests or assignments. These measures can provide insights into the spread of scores and help identify students who may need additional support.
Environmental Monitoring:

Range, variation, and standard deviation can be useful in environmental science and monitoring. For instance, in measuring air quality or water pollution levels over time, these measures can indicate the variability and trends in the data.
Business Process Improvement:

Organizations often use these measures to analyze process performance and identify areas where processes may be inconsistent or subject to variations. This is a key aspect of Six Sigma and Lean methodologies.
Sports and Athletics:

Range, variation, and standard deviation can be used to assess the performance of athletes, such as analyzing the spread of running times in a marathon or the consistency of scores in sports like gymnastics or figure skating.
Psychological and Social Sciences:

Researchers in psychology and social sciences use these measures to analyze survey data, experimental results, or behavioral patterns to understand variability in human behavior and attitudes.

In general, whenever you need to understand how data points deviate from a central value (such as the mean or median) and assess the spread or variability of data, range, variation, and standard deviation are valuable tools for analysis. They provide quantitative measures of dispersion that can inform decision-making and problem-solving in various fields.

Question 18

What is Chebyshev’s Theorem

Answer 18

Chebyshev’s Theorem
A theorem that (for any population) allows us to find an interval that contains a specified percentage of the individual measurements in the population

Question 19

How would you calculate Chebyshev’s Theorem on paper?

Answer 19

Question 20

How would you calculate Chebyshev’s Theorem on excel??

Answer 20

Question 21

On paper

Answer 21

Question 22

Solve using excel

Answer 22

Question 23

What is a stem-and-leaf display?

Answer 23

stem-and-leaf display
A graphical portrayal of a data set that shows the data set’s distribution by using stems consisting of leading digits and leaves consisting of trailing digits.

Question 24

How would you determine the first class interval from a stem and leaf display?

Answer 24

Determine the Interval Width:

The interval width is the difference between consecutive stem values. To determine the interval width, you can subtract the second stem value from the first stem value.

For example, if the first stem value is 10 and the second stem value is 20, the interval width would be 20 - 10 = 10.
Create the First Class Interval:

Use the first stem value and the interval width to create the first class interval.

The first class interval will start at the first stem value and end just before the next stem value.
For example, if the first stem value is 10 and the interval width is 10, the first class interval would be [10, 20).

Question 25

What is IQR?

Answer 25

interquartile range (denotedIQR)
The difference between the third quartile and the first quartile (that is,Q3−Q1)

Question 26

How do you find the median of a data set?

Answer 26

Question 27

What is the process for calculating percentile?

Answer 27

Question 28

How do you estimate the limits of a sample of the population

(Using the question pertaining to a collection of customer satisfaction ratings)

Answer 28