Chapter 3-4: Measures of Central Tendency & Intro to Variability

Question 1

If sample variance is computed by dividing SS by df = n – 1, then the average value of the sample variances from all the possible random samples will be _______ the population variance

a. smaller than
b. larger than
c. exactly equal to
d. unrelated to

Answer 1

c. exactly equal to

The answer is ‘exactly equal to’ because of a statistical concept known as Bessel’s correction. When you compute the sample variance by dividing the sum of squares (SS) by df = n - 1, it is referred to as the “unbiased” or “corrected” sample variance.

Bessel’s correction is used to make the sample variance an unbiased estimator of the population variance. By dividing by n - 1 instead of n (the population size), it accounts for the fact that you are estimating the population variance based on a sample, and it adjusts the calculation to provide a more accurate estimate.

When you take the average of these unbiased sample variances from all possible random samples, it is expected to be exactly equal to the population variance. This property is a fundamental concept in statistics and is a result of the mathematical adjustments made by Bessel’s correction to ensure that the sample variance is an unbiased estimator of the population variance.

Question 2

For a sample of n = 25 scores, how many scores are used to calculate the sample variance?

Answer 2

24

because for sample variance n-1 –> 25-1 = 24

Question 3

What measure of central tendency is appropriate for the measurement of something qualitative (eye color, diagnosis, etc)?

Answer 3

mode because mean and median are more appropriate for numerical/quantitative data

Question 4

On an exam with a mean of μ = 70, you have a score of X = 65. Which of the following values for the standard deviation would give you the highest position within the class?

a. σ = 1
b. σ = 5
c. σ = 10
d. cannot determine from the information given

Answer 4

c. σ = 10

If σ = 1, it means that the scores in the class are extremely tightly clustered around the mean (μ = 70), which would indicate that most students scored very close to 70. In this case, your score of 65 would still be significantly below the mean, and you might not achieve the highest position.

If σ = 5, it suggests that there is more variability in the scores, but it’s still relatively small. With a σ of 5, you have a better chance of having a higher position compared to a larger standard deviation, but it depends on the distribution of scores. If most scores are still clustered around 70, your score of 65 may not be enough for the highest position.

The choice of σ = 10 was suggested because it strikes a balance between having some variability in the scores (which increases your chance of having a higher position) while still keeping the scores relatively close to the mean. In this scenario, your score of 65 would likely be competitive for achieving a higher position within the class.

Ultimately, the specific answer depends on the actual distribution of scores in the class, but a smaller standard deviation generally increases your chances of having a higher position with a score below the mean.

Question 5

Scores from a statistics exam are reported as deviation scores. Which of the following deviation scores indicates a higher position in the class distribution

a. +8
b. 0
c. -8
d. cannot determine without more information

Answer 5

a. +8

In the context of deviation scores, these scores are calculated by subtracting the mean from an individual’s score. A positive deviation score means that the individual’s score is higher than the mean, while a negative deviation score means the individual’s score is lower than the mean.

So, a deviation score of +8 indicates that the individual’s score is 8 units above the mean, which implies a higher position in the class distribution. Conversely, a negative deviation score would indicate a lower position in the class distribution, as it would mean the individual’s score is below the mean.

Question 6

A distribution is positively skewed. Which is the most probable order, from smallest to largest value, for the three measures of central tendency?

Answer 6

Mode, Median, Mean

Question 7

A distribution is negatively skewed. Which is the most probable order, from smallest to largest value, for the three measures of central tendency?

Answer 7

Mean, Median, Mode

Question 8

On an exam with a mean of μ = 70, you have a score of X = 75. Which of the following values for the standard deviation would give you the highest position within the class?

a. σ = 1
b. σ = 5
c. σ = 10
d. cannot determine from the information given

Answer 8

a. σ = 1

Question 9

If you have a score above the mean, you would want the SD to be smaller, to give you the highest position in the class.

True or False

Answer 9

true

Question 10

If you have a score below the mean, you would want the SD to be smaller, giving you less variability, to give you the highest position in the class.

True or False

Answer 10

False, you would want the SD to be larger, giving you more variability to make it appear as though you have a higher position in the class

Question 11

A population with a mean of μ= 8 and has a ΣX= 56. How many scores are in the population?

Answer 11

Mean = ΣX/N

So to calculate the scores (N):

N=ΣX/ μ
N= 56/8
N= 7

Question 12

A population with a mean of μ= 8 and has a N= 5. What is the total number of scores (ΣX) in the population?

Answer 12

Mean (μ) = ΣX/N

So to calculate the ΣX:

ΣX= μ x N
ΣX = 8 x 5
ΣX = 40

To calculate the total number of scores (ΣX), you would need to know the actual scores of the 5 individuals in the population. So, technically this cannot be calculated

Question 13

For a negatively skewed distribution with a mean of M = 30, what is the value of the median?

a. greater than 30
b. less than 30
c. 30
d. cannot be determined from the info given

Answer 13

a. greater than 30

because from smallest to largest for a negatively skewed distribution, it goes mean, med mode

Question 14

How many of the scores are used to calculate the range if the sample consists of n =43

Answer 14

only 2 numbers; the highest number and the lowest number are needed to calculate the range

Question 15

What are the steps for computing the SD (or s) ?

Answer 15

1. Calculate the variance (SS/N)
2. Take the squared root

Question 16

What formula is this?

s ² = (Σ X ² - (Σ X)²/N ) /N

Answer 16

Computational formula to calculate the variance

Question 17

What are the steps for computing the variance using the definitional formula?

5pts

Answer 17

1. Calculate the mean (add all the scores and divide by the number of scores)
2. Subtract the mean from each score (X-M)
3. Square each of these deviation scores (X – M)²
4. Add up the squared deviation Σ (X – M) ² (SS)
5. Divide the sum of squared (SS) deviation scores by the number of scores Σ (X – M) ² /N or SS/N

Question 18

Define:

Most common way of describing the spread of a group of scores

Answer 18

Standard Deviation

Question 19

Name:

Is the single number that represents the total amount of variability in the distribution.

The larger this number is, the greater the total spread of scores

Answer 19

The variance (s², σ²)

Question 20

What do these symbols represent?

s², σ²

Answer 20

s² = variance of a sample
σ² = variance of a population

Question 20

What is the mean? What is the symbol for a sample vs population?

Answer 20

The average. Add up all the scores in the distribution and divide by the number of scores

M (sample) & µ (population): called Mu

Question 21

What is the median?

Answer 21

Frist rank order the scores and then the one that divides the distributions into equal halves is the median. It is the point at which half of the scores are below and half of the scores are above

Question 22

What is the mode?

Answer 22

Is the number or evet that occurs the most frequently in a distribution

Question 23

What are the 3 types of measures of central tendency?

Answer 23

mean, median and mode

Question 24

True or false

The median is affected by extreme scores.

Answer 24

False- it is unaffected by extreme scores

Question 25

Fill in the blank:

BLANK balances the number of scores

Answer 25

Median

Question 26

Fill in the blank:

BLANK balances the distance of scores

Answer 26

Mean

Question 27

What does the formula Σ(X-M) = 0
signify ?

Answer 27

The distance above and below the mean will always cancel out and sum up to 0 because they are both the same number just diff sign (pos or neg)

Question 28

Define:

What is the deviation score?

Answer 28

(X-M) –> should always sum up to 0

Question 29

Name:

How close the scores are to the mean or how far apart/spread out they are from the mean

Answer 29

Variation

Question 30

What is it called when you add up all the deviation scores and divide by the number of scores? What is the result?

Answer 30

-It is called calculating the average deviation
-The result is you get zero

Question 31

What happens to the mean if you remove a score (x) lower than the mean (M) from a sample?

Answer 31

The mean will increase

Question 32

When there is a distribution table involved, how would you calculate the mean?

Answer 32

You would multiply the sum of x and the sum of f divided by the sum of f (sum of frequencies)

Mean= Σ(x * f) / Σf

Also, Σf = N –> Because it is the total number of observations which is also the sum of frequencies

Question 33

If you add a constant to the X values, the mean will also change by that constant.

True or false

Answer 33

True

Question 34

Fill in the blanks:

In situations where you have missing or undetermined values or where outliers may distort the BLANK significantly, the BLANK is often a better choice as a measure of central tendency because it is more robust and less sensitive to BLANK values.

Answer 34

• Mean
• Median
  -Extreme

Question 35

A researcher is measuring problem-solving times for a sample of n= 5 20 laboratory
rats. However, one of the rats fails to solve the problem so the researcher has an
undetermined score. What is the best measure of central tendency for these data?

Answer 35

The median

Question 36

What is the best measure of central tendency for an extremely skewed
distribution of scores?

Answer 36

The median

Question 37

Which of the following is a consequence of increasing variability?

a. The distance from one score to another tends to increase and a single score tends to provide a more accurate representation of the entire distribution.

b. The distance from one score to another tends to increase and a single score tends to provide a less accurate representation of the entire distribution.

c. The distance from one score to another tends to decrease and a single score tends to provide a more accurate representation of the entire distribution.

d. The distance from one score to another tends to decrease and a single score
tends to provide a less accurate representation of the entire distribution.

Answer 37

b. The distance from one score to another tends to increase and a single score tends to provide a less accurate representation of the entire distribution.