# Central tendency

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Central tendency

## 1. Mean

### 1.1. Serves as a balance point

1.1.1. Always located inside the group of scores

### 1.2. Sum of the scores divided by the number of scores.

1.2.1. How to calculate means

1.2.1.1. Population: μ = (Σ * X)/ N

1.2.1.2. Sample: x̄ = ( Σ xi ) / n

### 1.3. Every score in the distribution contributes to the value of the mean

1.3.1. Changing the value of any score will change the mean

## 2. Mode

### 2.3. It is possible to have more than one.

2.3.1. A distribution with two modes is said to be bimodal

2.3.2. A distribution with more than two modes is called multimodal.

### 2.5. When to use the Mode

2.5.1. Nominal Scales

2.5.2. Discrete Variables

2.5.3. Describing Shape

## 3. There are three measures of central tendency

### 3.1. Median

3.1.1. Serves as the midpoint

3.1.2. The goal is to locate the midpoint of the distribution.

3.1.3. No symbols or special notation

3.1.4. When to Use the Median

3.1.4.1. Extreme Scores or Skewed Distributions

3.1.4.2. Undetermined Values

3.1.4.3. Open-Ended Distributions

3.1.4.4. Ordinal Scale

## 4. z = (x – μ) / σ

### 4.2. standardizing a distribution to create new values is a two-step process:

4.2.1. The original raw scores are transformed into z-scores.

4.2.2. The z-scores are then transformed into new X values so that the specific μ and standard deviation are attained.

## 6. Variability

### 6.2. Range

6.2.1. Distance covered by the scores in a distribution, from the smallest score to the largest score.

### 6.3. Variance

6.3.1. Equals the mean of the squared deviations. Variance is the average squared distance from the mean.

### 6.4. Deviation

6.4.1. Distance from the mean

### 6.5. Standard deviation

6.5.1. The square root of the variance and provides a measure of the standard distance from the mean.

## 7. Variance

### 7.1. Population variance

7.1.1. Represented by the symbol and equals the mean squared distance from the mean. Population variance is obtained by dividing the sum of squares by N.

### 7.2. Population standard deviation

7.2.1. Represented by the symbol and equals the square root of the population variance.

### 7.3. Sample variance

7.3.1. Represented by the symbol and equals the mean squared distance from the mean. Sample variance is obtained by dividing the sum of squares by n − 1.

### 7.4. Sample standard deviation

7.4.1. Represented by the symbol s and equal the square root of the sample variance.

## 8. Z-scores

### 8.2. If every X value is transformed into a z-score, the distribution of z-scores will have:

8.2.1. Shape

8.2.2. The mean

8.2.3. Standard deviation