The normal distribution, standard deviation and SEM

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Garrick Shelton

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33 questions

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Open Ended

30 sec

1 pt

Statistical symbols

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Answer explanation

Open Ended

30 sec

1 pt

What is the sample size n?

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Answer explanation

The number of values x in a sample.

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30 sec

1 pt

What is the main logic when remembering the symbols for sample and population parameters?

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Answer explanation

The sample symbols are English letters, the population are in Greek (also, n is lower case for samples and uppercase for population).

Open Ended

30 sec

1 pt

What are the symbols for variance?

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σ² and s²

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30 sec

1 pt

What is the population variance?

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Answer explanation

The mean of the squares, minus the square of the mean. It is the average distance of the squared data points from the mean and has arbitrary units.

Open Ended

30 sec

1 pt

What is the sample variance?

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Answer explanation

An estimate of σ² that uses a representative sample, rather than the whole population.

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30 sec

1 pt

Why do formulae for sample variance and sample standard deviation contain n - 1 rather than n in the denominator?

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Answer explanation

To give more conservative estimates of their respective magnitudes, by increasing their ratio.

Open Ended

30 sec

1 pt

What is the population standard deviation?

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The square root of the population variance. It is the average distance of the data points from the mean and has units of the original data.

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30 sec

1 pt

What is the sample standard deviation?

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An estimate of σ the uses a representative sample, rather than the whole population.

10.

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30 sec

1 pt

How are the standard deviation and the variance related?

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Answer explanation

The standard deviation is the square root of the variance (σ = √σ²).

11.

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30 sec

1 pt

What is the normal distribution?

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Answer explanation

The distribution that describes a data set where 68% of values are within 1 S.D, 95% of values are within 2 S.D and 99% of values are within 3 S.D of the mean.

12.

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30 sec

1 pt

What is the x-axis of the bell-shaped curve?

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Answer explanation

It shows the mean (centre) and the 3 associated positive and negative standard deviations.

13.

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30 sec

1 pt

What is the statistical definition of the general normal distribution?

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Answer explanation

X ~ N(µ, σ²) (the continuous random variable X follows a normal distribution with mean µ and variance σ²).

14.

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30 sec

1 pt

What is the statistical definition of the standard normal distribution?

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Z ~ N(0, 1²) (the continuous random variable Z follows a normal distribution with mean 0 and variance 1²).

15.

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30 sec

1 pt

How is a measured random variable X (general normal) converted into the random variable Z (standard normal)?

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Answer explanation

Z = (X - µ) ÷ σ (population) or Z = (X - x̄) ÷ s (sample) or use SEM when calculating z-scores for sampling distributions

16.

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30 sec

1 pt

What is the Z score more intuitively?

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Answer explanation

The factor that you need to multiply the sample standard deviation by to obtain a random variable X (normalises X with standard deviation). Dividing by SEM tells how many average distances away from the mean a value is, which if the data is normally distributed follows the bell-shaped curve.

17.

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30 sec

1 pt

What is the relationship between probability and the standard normal bell-shaped curve?

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Answer explanation

The probability of the random variable X being less than or greater than x standard deviations from the mean is given by the area under the curve to the left or right of x, respectively.

18.

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30 sec

1 pt

How is the probability of obtaining a value greater or less than that of an observed random variable X calculated?

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Answer explanation

X is first converted into the Z score. The P(Zz) is then read from the Z-scores table.

19.

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30 sec

1 pt

What is a sampling distribution?

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Answer explanation

The probability distribution of multiple sample means. So each time, you take n random samples of a population (some examples use repeated random sampling from a small population) and take their mean. A sampling distribution is then created, which becomes normally distributed as n is increased.

20.

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30 sec

1 pt

What is the standard deviation of a sampling distribution known as?

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The standard error of the mean (SEM).

21.

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30 sec

1 pt

What is the formula for the SEM?

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Answer explanation

The standard deviation of the sampling distribution divided by the square root of n.

22.

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30 sec

1 pt

For large sample sizes, what are sampling parameters equivalent to?

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Answer explanation

Population parameters.

23.

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30 sec

1 pt

What are confidence intervals?

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Answer explanation

The range of values that X% of the sample means (or other RV) lie within.

24.

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30 sec

1 pt

What is the general word formula for the confidence interval?

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Answer explanation

Point estimate ± Z*error (point estimate) Where the point estimate is often the mean, Z the z-score of the confidence limit (e.g. 95%) and error the SD.

25.

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30 sec

1 pt

What is the formula for the confidence interval for a sample mean?

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Answer explanation

x̄ ± z*(s/√n)

26.

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30 sec

1 pt

What is the expected value E(X) of a random variable?

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Answer explanation

It is the probability-weighted average of all of its possible values. All of the possible values of X are multiplied by their probability of occurring and the values summed.

27.

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30 sec

1 pt

How is the 95% confidence interval calculated when the population SD is known?

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Answer explanation

First, the population mean is inferred from the sampling distribution, and the SEM calculated. The z scores which are above and below 95% of the sample means are 1.96 and -1.96. These are multiplied by the SEM to give the distance from the mean that 95% of values fall within. Remember that it is the fact that the data is normally distributed that enables the CI to be calculated. How good those CI are is mainly dependent on the sample size, as SEM decreases with SQRT(n).

28.

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30 sec

1 pt

Why does increasing n decrease the SEM to one which is smaller than the original population SD?

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Answer explanation

Because increasing the number of samples provides a better estimate of the mean of the population, with a decreased SAMPLING error.

29.

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30 sec

1 pt

What is the students t distriubtion?

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Answer explanation

An alternative distribution which is used to approximate probabilities when the population SD is unknown, and sample SD is used instead. Students t distribution changes with sample size and the degrees of freedom.

30.

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30 sec

1 pt

How is the 95% confidence interval calculated when population SD is unknown?

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Answer explanation

Rather than using z scores, t scores are used for the alpha values. As the t distribution changes with sample size, so do the associated values of alpha. These are then multiplied by the sample SEM (s/SQRT(n)), which is an approximation of the population SD.

31.

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30 sec

1 pt

What are the characteristics of the students t distribution?

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Answer explanation

1. Shorter in the middle, fatter in the tails 2. There is a t distribution for every sample size 3. Degrees of freedom n-1 4. Smaller sample sizes result in distributions that are shorter and fatter with more tail probability 5. As n becomes large, the t distribution converges on the z distribution 6. All sample means within the t distribution are the same size

32.

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30 sec

1 pt

When calculating CI when population SD is unknown, do samples of different sizes have the same SEM?

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Answer explanation

No, because different sample SDs are used in the numerator, changing sample size changes the corresponding SEM.

33.

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30 sec

1 pt

What is the central limit theorem?

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Answer explanation

If I take multiple random samples of size n from a population, as the number of times I sample the population goes to infinity the distribution of the means of these samples approach a normal distribution with a mean that approximates the population mean. Because intersample variability is lower when the sample size n is higher, SEM with high n give a more precise estimate of the variability in the sampling distribution, and more confidence that the sample statistic accurately reflects the population parameter (e.g. the population mean). This is shown with confidence intervals, where SEM is multiplied by the z scores to show the difference between the sample mean and the population mean to show the range of values within which 95% of the observed means fall within.

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