CENTRAL TENDENCY VARIABILITY

Question 1

Do frequency distribution allow quantitative statements between distributions?

Answer 1

No, observations and counts are not enough to make measurable comparisons. Central tendencies (means) and dispersions of scores (variability) are best because most stats tests require them and they can be useful ont their own.

Question 2

What is the definition of arithmetic mean?

Answer 2

The mean is a measure of central tendency. It is the sum of the scores divided by the number of scores.

It is the best single number for describing a group of scores; it gives the central tendency, or the typical value of a group of scores

Question 3

When can you calculate the mean of a population?

Answer 3

If you have access to all the scores of a population of interest, then the mean of that population can be represented by μ (mu).

Question 4

Is the mean of a population set of scores a parameter?

Answer 4

Yes

Question 5

What are the five properties of the mean?

Answer 5

1. The mean is sensitive to the exact value of all scores in the distribution: you change one value, the mean changes.
2. The sum of the deviations about the mean equals zero. Written algebraically this
   property becomes:
   ∑(Xi −MEAN(X barre en haut))=0

*Deviation means a quel point un score dévie/s’éloigne de la moyenne

That is why we say the mean is the balancing point in a distribution; values over and under mean cancel each other out.

3. The mean is sensitive to extreme scores.

4.The sum of the squared deviations of all the scores about their mean is a minimum.
Stated algebraically,
∑(Xi − X)2 is a minimum.

5.Under most circumstances, out of all the measures used for central tendency (e.g. median), the mean is the least subject to sampling variation (differences between samples of a population). Dans le sens que les differences moyennes will not be THAT different from one another.

Question 6

What is a weighted mean?

Answer 6

The weighted mean is a type of average that gives different levels of importance (or “weight”) to different groups or data points. It ensures that larger or more important groups have a bigger influence on the overall average than smaller or less important ones.

two ways: when raw scores are available and when they aren’t (mean each group multiplied by number of scores in that group…)

Question 7

If you have 3 groups: group 1 has two scores, group 2, 3 scores and group 3, 5 scores. Can you sum up all their means and divide by the number of groups to give the overall mean?

Answer 7

No. You can only do this when the sample sizes (number of scores) are the same from group to group. When that is not the case you have to take into account the weight of each sample group.

Question 8

What is the median?

Answer 8

The scale value below which 50% of the scores fall. In other words, it is the
middle score of a distribution.

Question 9

What is the median in ungrouped odd and even scores?

Answer 9

In ungrouped scores (which is most often encountered) the median is the centermost score if the number of scores is odd. If the number of scores is even, the median is taken as the average of the two centermost scores.

Question 10

What is an important property of the median?

Answer 10

The median is not affected by extreme scores.

Question 11

What is the mode ?

Answer 11

the most frequent score in the distribution.

Question 12

What is a unimodal distribution versus a bimodal distribution?

Answer 12

Unimodal: there is only one score that most frequently stands out.

Bimodal: distributions that has two modes (two common values)

keep in mind, possible for more….

Question 13

When is the mode most useful as a central tendency as a measure of central tendency?

Answer 13

a) there are relatively few values
b) the values are categories (ex: blue, green, blue, yellow…) - (not quantitative date - so you cat calculate the mean or find the median)

Question 14

When is the mean = median = mode?

Answer 14

When the distribution is unimodal and symmetrical (bell-shaped curve)

Question 15

Where can you find the median in a positive or negative skew?

Answer 15

Between the mode and mean.

Question 16

When is the mean never equal to the median?

Answer 16

When the distribution is skewed.

Question 17

What do measures of variability tell us?

Answer 17

How spread out the scores are in a distribution.

Question 18

What are 4 measures of variability (comment tes scores varient)?

Answer 18

Range, deviation score, variance, standard deviation.

Question 19

How do we calculate the range?

Answer 19

The difference between the highest and lowest score in the distribution

Range = (highest score – lowest score) + smallest measuring unit

example:
c. 1.2, 1.3, 1.5, 1.8, 2.3
Ranges = (2.3 – 1.2) + .1 = 1.2

Question 20

What does the deviation score tell us?

Answer 20

It tells how far away the raw score is from the mean of the distribution.

X-Mean. Remember,
sum (X-Mean) always = 0

Question 21

What is the variance?

Answer 21

The average of the squared distances of all the values from the mean.

Def des notes: an index that reflects the degree of variability in a group of scores. It does not have the limitations of the range

It is used only occasionally as a descriptive statistic since it is scaled in squared units.

Question 22

What is the difference between a sample variance and a population variance?

Answer 22

If you have access to all the scores of a population of interest, then the variance can be represented by o2 and not s2. The sum of squares (SS) is also divided by N rather than n-1 (Degrees of freedom (df)). This is because we are not estimating anymore, we are not leaving any room for the rest of the population.

Question 23

The sample mean estimates….

Answer 23

The population mean

Question 24

The sample variance estimates…

Answer 24

The population variance

Question 25

le u = population mean. Hoe else can we call it?

Answer 25

The true mean

Question 26

What is the pooled variance?

Answer 26

The average variance of several groups. Beware, just like the weighted mean, we can't sum up the variances form each group and then divide that number by the number of groups we have. We have to take into account the size of each group. We can only do that if the sizes of the groups are the same (le nombre de valeurs est le meme). Makes sense perch écoute si ta plus de valeurs dans un groupe, eh bien son average est surement plus representative, alors faut tenir ca en compte, porch il reste que le but est de estimate the population mean!!!!!!! In other words, making the pooled variance a more accurate reflection of the overall variability across all groups. Imagine you're comparing two groups: Group 1: The heights of 5 kids in one class. Group 2: The heights of 5 kids in another class. Now, you want to compare the variability (how spread out the heights are) in these two groups.

Question 27

Can we calculate the pooled variance of a population?

Answer 27

No because pooled variances are used when we have many samples of one population. When you have a population (meaning you know the full set of data points), there is no need for pooling, because you already have all the data and can calculate the variance directly from the population itself.

Question 28

Ca c'est juste un example pour que tu comprennes mieux les groupes writhin a population and the need of pooled variance:

Answer 28

Example 2: Comparing Treatment Effectiveness in Different Age Groups In another study, you might want to compare the effectiveness of a new depression treatment across two age groups: Group 1 (Young Adults): 20 young adults aged 18-30 who receive the treatment. Group 2 (Older Adults): 20 older adults aged 60-75 who receive the treatment. Again, both groups come from the same population (adults who need treatment for depression). You calculate the variance in how much depression scores improved in each group. Group 1 (Young Adults): Calculate the variance of depression score changes in this group. Group 2 (Older Adults): Calculate the variance of depression score changes in this group. Now, you want to get a pooled variance to estimate the overall variability of the treatment’s effect across both age groups, and this would give you a better, more combined picture of how the treatment works for people in both groups.

Question 29

What is the standard deviation?

Answer 29

It is the square root of the variance

Question 30

Why is the standard deviation a better measure of variability than variance?

Answer 30

Because it is in the original unit of measurement (not squared).

Question 31

What are three ways the variance can be calculated?

Answer 31

Variance of the sample, variance of the population and variance of several groups (samples).

Question 32

What does the coefficient of variation indicate? How do we calculate it?

Answer 32

The coefficient of variation (CV) is like a percentage that tells you how big the standard deviation is compared to the mean (average). Formula: variance/mean x 100.

Question 33

What does the variance tell us? What does the standard deviation tell us?

Answer 33

Variance = 8: This tells us about the overall spread of the data, but it’s in squared units, which is harder to interpret. Standard Deviation ≈ 2.83: This is the average distance of each number from the mean (in the same units as the data, which makes it easier to understand). So, the data points (2, 4, 6, 8, 10) are, on average, about 2.83 units away from the mean (6).

Question 34

What are the 6 properties of s2 (variance) and s (standard deviation)?

Answer 34

1.Sum of the squared deviations about the mean is less than about any other value (property of the mean remember!!) 2.Numbers are always positive because squaring of negative numbers results in positive values 3. A large deviation from the mean contributes disproportionately to the total (4 squared =16 but 8 squared= 64) 4. As the variability in the distribution increases, the statistical variance also increases. (plus loin du mean, makes sense) 5. If all scores are in distribution are identical, the variance will equal zero. (Identical to the mean makes sense) 6.Under certain conditions, the variance can be partitioned and its portions attributed to different sources: MEANING: This statement means that the total variance in a dataset can be broken down into parts (or "portions") that come from different causes or sources. Partitioning variance helps researchers and analysts understand what’s causing variability in their data. For example: In psychology, you might study how much variance in happiness is due to genetics vs. life circumstances.

Question 35

The smaller the standard deviation is (and variance), the...

Answer 35

closer the numbers are from here eachother

Question 36

For your example: Mean = 50 kg SD = 5 kg It tells us that most weights in the dataset are within 5 kg above or below the mean (so between 45 kg and 55 kg, roughly). It’s a good estimate of how far the weights typically vary from the mean. CV= 10% This means the standard deviation (5 kg) is 10% of the mean (50 kg). It gives a sense of how large the variation is relative to the average weight. This is useful when comparing datasets with different scales or units.

Answer 36

xxx