7 Measures of Central Tendency and Dispersion

Question 1

What is a distribution in statistics?

Answer 1

A function that describes the probabilities of any theoretical outcome based on evidence from a study.

Distributions help in understanding how likely a specific value is when compared to historical data.

Question 2

What is the purpose of creating a histogram?

Answer 2

To visualize the distribution of data and understand its shape.

A histogram allows analysts to see the frequency of different values in a dataset.

Question 3

What is a normal distribution?

Answer 3

A bell-shaped distribution where the mean, median, and mode are all at the center, and probabilities decrease evenly as you move away from the center.

It is one of the most common distributions in statistics.

Question 4

What does a uniform distribution look like?

Answer 4

A flat line where every value has the same probability of occurring.

In a uniform distribution, there is no variability among the data points.

Question 5

What is a Poisson distribution?

Answer 5

A distribution that describes the number of times an event occurs in a fixed interval of time or space.

Examples include counting occurrences of events like website hits or roadkill.

Question 6

What is the main characteristic of an exponential distribution?

Answer 6

It describes the time between events in a Poisson process, typically showing a sharp curve.

Exponential distributions can indicate rapid changes in a variable.

Question 7

What are Bernoulli distributions?

Answer 7

Distributions with only two possible outcomes, such as true/false or success/failure.

They are used to model binary variables.

Question 8

What is the key difference between Bernoulli and binomial distributions?

Answer 8

Bernoulli distributions have one trial, while binomial distributions involve multiple trials.

Binomial distributions calculate the probability of a specific number of successes across several trials.

Question 9

What does skew measure in a distribution?

Answer 9

The degree of asymmetry of a distribution around its mean.

A distribution can be negatively skewed (left) or positively skewed (right).

Question 10

What is kurtosis?

Answer 10

A measure of the ‘tailedness’ of a distribution, indicating how much of the variance is due to extreme values.

Distributions can be leptokurtic (tall and skinny) or platykurtic (short and wide).

Question 11

What are the three measures of central tendency?

Answer 11

• Mean
• Median
• Mode

These measures summarize a dataset with a single value representing the center.

Question 12

How is the mean calculated?

Answer 12

By summing all values and dividing by the number of values.

This represents the average of the dataset.

Question 13

What is the median in a dataset?

Answer 13

The middle value when the data is ordered from least to greatest.

If there is an even number of values, the median is the average of the two middle numbers.

Question 14

What is the mode?

Answer 14

The value that appears most frequently in a dataset.

A dataset can have more than one mode or no mode at all.

Question 15

What is the mean?

Answer 15

The mean is also called the average. To find the average, you take the sum of the values and divide them by the number of values.

Question 16

What are the steps to calculate the mean?

Answer 16

• Find the sum of the values.
• Divide the sum by the number of values.

Question 17

Calculate the mean of the weights: 22 lb, 26 lb, 24 lb.

Answer 17

Mean = 24 lb

Sum = 22 + 26 + 24 = 72; Mean = 72 / 3 = 24

Question 18

What is the median?

Answer 18

The median is the middle value of a dataset when arranged in ascending order.

Question 19

What are the steps to find the median?

Answer 19

• Arrange values in ascending or descending order.
• Find the value that is in the exact middle.
• If the dataset has an even number of values, find the mean of the two middle values.

Question 20

Find the median of the weights: 22 lb, 24 lb, 26 lb.

Answer 20

Median = 24 lb

Middle value of 22, 24, 26 is 24.

Question 21

Find the median of the weights: 22 lb, 24 lb, 25 lb, 26 lb.

Answer 21

Median = 24.5 lb

Middle values are 24 and 25; (24 + 25) / 2 = 24.5.

Question 22

What is the mode?

Answer 22

The mode is the number that occurs most often in a dataset.

Question 23

What are the steps to find the mode?

Answer 23

• Arrange values in ascending or descending order.
• Count the occurrences of all repeating values.
• Compare the number of occurrences for every repeated value.

Question 24

What is the mode of the weights: 22 lb, 25 lb, 25 lb, 24 lb, 24 lb, 23 lb, 25 lb, 22 lb?

Answer 24

Mode = 25 lb

25 occurs most frequently.

Question 25

True or False: The mode can have no value if no number is repeated.

Answer 25

True.

Question 26

When should you use the mean?

Answer 26

Use the mean as a default; it works well with normal distributions.

Question 27

When is the median preferred over the mean?

Answer 27

Use the median when dealing with skewed or asymmetrical data.

Question 28

When is the mode useful?

Answer 28

The mode is ideal for handling nominal variables.

Question 29

What is range?

Answer 29

Range is the difference between the maximum and minimum values in a dataset.

Question 30

What are the steps to calculate the range?

Answer 30

* Arrange values in ascending or descending order. * Identify the minimum and maximum values. * Subtract the minimum from the maximum.

Question 31

Calculate the range of the following dataset: 12, 18, 10, 22, 15, 25, 16, 17, 14, 19.

Answer 31

Range = 15 ## Footnote Max = 25, Min = 10; 25 - 10 = 15.

Question 32

What are quartiles?

Answer 32

Quartiles divide your data into four equal parts.

Question 33

What are the steps to find quartiles?

Answer 33

* Arrange values in ascending or descending order. * Find the median (Q2). * Split the dataset at the median. * Find the median of the lower dataset (Q1) and the upper dataset (Q3).

Question 34

Find the quartiles of the dataset: 12, 18, 10, 22, 15, 25, 16, 17, 14, 19.

Answer 34

Q1 = 14, Q2 = 16.5, Q3 = 19 ## Footnote Q2 is the median; Q1 is median of lower half; Q3 is median of upper half.

Question 35

What is the interquartile range?

Answer 35

The interquartile range is the difference between Q3 and Q1.

Question 36

Calculate the interquartile range given Q1 = 14 and Q3 = 19.

Answer 36

Interquartile range = 5 ## Footnote 19 - 14 = 5.

Question 37

What is variance?

Answer 37

Variance is a measure of dispersion that looks at the squared deviation of each data point from the mean.

Question 38

What are the steps to calculate variance?

Answer 38

* Find the mean of the dataset. * Subtract the mean from each data point. * Square the results. * Find the sum of the squared results. * Divide by the number of data points minus 1 (for samples).

Question 39

Calculate the variance for the dataset: 3, 6, 4, 7, 5, 1, 9, 5, 4, 6.

Answer 39

Variance = 4.9 ## Footnote Mean = 5; Squared deviations summed = 44; 44 / (10 - 1) = 4.9.

Question 40

What is standard deviation?

Answer 40

Standard deviation is the square root of the variance.

Question 41

Why is standard deviation important?

Answer 41

It provides a measure of dispersion in the same units as your dataset.

Question 42

What happens when you apply standard deviation to a normal distribution?

Answer 42

One standard deviation above the mean includes 34.1% of your data points.

Question 43

What is the square root of the variance called?

Answer 43

Standard deviation

Question 44

Why is standard deviation important?

Answer 44

It puts the standard deviation in the same units as your dataset

Question 45

What percentage of data points are within one standard deviation of the mean in a normal distribution?

Answer 45

68.2%

Question 46

What percentage of data points are within two standard deviations of the mean in a normal distribution?

Answer 46

95.4%

Question 47

What percentage of data points are within three standard deviations of the mean in a normal distribution?

Answer 47

99.6%

Question 48

What is a common cutoff for outliers in data analysis?

Answer 48

Three standard deviations

Question 49

What is the first step in calculating standard deviation?

Answer 49

Find the mean of the dataset

Question 50

What do you do after finding the mean in the standard deviation calculation?

Answer 50

Subtract the mean from each data point

Question 51

What do you do after subtracting the mean from each data point?

Answer 51

Square the results

Question 52

What is done after squaring the results in the standard deviation calculation?

Answer 52

Find the sum of the squared results

Question 53

What do you divide the sum of squared results by?

Answer 53

Number of data points minus 1

Question 54

What do you take the square root of in the standard deviation calculation?

Answer 54

Variance

Question 55

What is the mean of the dataset: 4, 5, 3, 4, 2, 2, 6, 4, 6, 4?

Answer 55

4

Question 56

What is the variance of the dataset: 4, 5, 3, 4, 2, 2, 6, 4, 6, 4?

Answer 56

2

Question 57

What is the standard deviation of the dataset: 4, 5, 3, 4, 2, 2, 6, 4, 6, 4?

Answer 57

1.41

Question 58

What are common types of distributions?

Answer 58

* Normal * Uniform * Poisson * Exponential * Bernoulli * Binomial

Question 59

What is skew in data distribution?

Answer 59

How data is distorted left or right

Question 60

What is kurtosis in data distribution?

Answer 60

How data is distorted up or down

Question 61

What are the common measures of central tendency?

Answer 61

* Mean * Median * Mode

Question 62

What are the simpler methods of measures of dispersion?

Answer 62

* Range * Quartiles

Question 63

What are more complicated methods of measures of dispersion?

Answer 63

* Variance * Standard deviation

Question 64

What does standard deviation measure?

Answer 64

The average distance of every point from the mean

Question 65

True or False: The middle quartile (Q2) is the same as the median.

Answer 65

True

Question 66

What is the mode of the dataset: 24, 18, 36, 51, 24, 48, 18?

Answer 66

* 18 * 24

Question 67

What is the range of the dataset: 15, 615, 46, 73, 45, 80, 46?

Answer 67

600

Question 68

What is the middle quartile (Q2) of the dataset: 10, 24, 13, 9, 15, 7, 19?

Answer 68

13

Question 69

What is the standard deviation of the sample dataset: 9, 11, 7, 8, 9, 10?

Answer 69

√2