Lecture 5 - The normal distribution and z-scores

Question 1

Define mode, median and mean

Answer 1

• mode: the value of the most commonly occurring score
• median: the value that half of the scores fall below - the 50th percentile
• mean: the value from which deviation scores sum to zero, found by summing the scores and then dividing by the number of scores

Question 2

describe the shapes of different distributions

Answer 2

• normal distribution: symmetrical, bell-shaped curve
• bimodal distribution: two peaks
• positively skewed distribution: tail extends to the right
• negatively skewed distribution: tail extends to the left

Question 3

why is variability important in research?

Answer 3

variability is important because it indicates the spread of scores in a dataset. It helps in understanding the reliability of sample means compared to the population mean, and it allows estimation of how much a sample mean will likely differ from a population mean based on sample size and population variability

Question 4

what are the two major sources of variability?

Answer 4

the two major sources of variability are stable individual differences (traits of participants) and situation factors (eg. background noise during a testing session, participants’ experiences on the day of testing)

Question 5

what are the steps to calculate the standard deviation?

Answer 5

1. add up all the scores and divide by the number of scores to get the mean
2. subtract the mean from each score to get the deviation scores
3. square each deviation score
4. sum the squared deviation scores
5. divide the sum of squares by the number of scores to get the variance
6. take the square root of the variance to get the standard deviation

Question 6

what is a frequency distribution?

Answer 6

a frequency distribution is a summary of how often each value occurs in a dataset. It can be represented graphically, often in the form of a histogram

Question 7

How is a z-score calculated?

Answer 7

A z-score is calculated using the formula attached
where X is the raw score μ is the mean and σ is the standard deviation

Question 8

convert a raw score of 110 to a z-score given a mean of 100 and a standard deviation of 10

Answer 8

see photo for answer

Question 9

what are the characteristics of the normal distribution?

Answer 9

the normal distribution is a bell-shaped curve that is symmetrical around the mean
it describes the distribution of many variables, and its properties are used in many inferential statistical tests. The mean, median, and mode of a normal distribution are all equal

Question 10

describe the percentage of scores within one, two and three standard deviations of the mean in a normal distribution

Answer 10

• within one standard deviation: 68%
• within two standard deviations: 95%
• within three standard deviations: 99%

Question 11

How do you convert a z-score to a percentile?

Answer 11

converting a z-score to a percentile involves using z-tables which provide the percentage of scores that lie below a given z-score in a standard normal distribution

Question 12

what percentage of runners finish a marathon in under 5.5 hours if the mean finishing time is 4.5 hours with a standard deviation of 1 hour?

Answer 12

using a z-table, a z-score of 1 corresponds to the 84th percentile, meaning 84% of runners finish in under 5.5 hours

Question 13

Given Marissa’s music theory score of 65/00 and practical score of 75/100, with averages and standard deviations of 50 and 10 for theory, and 60 and 15 for practical, respectively, on which exam did she do better?

Answer 13

Marissa did better on the theory exam with a z-score of 1.5 compared to 1 for the practical exam

Question 14

convert the following z-scores to raw scores given the means and standard deviations:
- maths: z = 1.5, μ = 55, σ = 3
- science: z = -0.4, μ = 60, σ = 10
- english: z = -1.2, μ = 80, σ = 5

Answer 14

• maths: X = (1.5×3)+55=59.5
• science: X = (−0.4×10)+60=56
• english: X = (−1.2×5)+80=74