LESSON 8

LESSON 8 MEASURES OF CENTRAL TREND, POSITION AND DISPERSION.

They apply only to quantitative variables.

1.  NUMERICAL SUMMARY OF A STATISTICAL SERIES.

There are three major types of statistical measures:

- Measures of position: idea of ​​the magnitude, size of the position of observations of the data once they are ordered from minor to a mayor.

- Measures of central tendency: idea of ​​the central behavior of the subjects.

- Measures of dispersion or variability: information about the heterogeneity of subjects, ie whether they are very different from each other or not.

They only apply to continuous quantitative variables.

2. CENTRAL TREND MEASURES

• Arithmetic mean or mean: for quantitative variables.

It is calculated for quantitative variables and is the geometric or gravity center of our data.

When the data are grouped (two intervals), to calculate the mean we use as reference value of each interval its class mark: we calculate a weighted arithmetic mean that is obtained by summing the class mark by the absolute frequency, between N.

• x= Ʃm_c (mark of class) f_i /n (We multiply the mark of class by the absolute frequency and we are adding, then we divide between the number of subjects)
• Median: Value that occupies the center position in an ordered list of data.

• Property: robustness. It only takes into account the position of the values ​​in the sample and therefore has much better behavior than the average when there are extreme observations.

• Mode: It is the value most frequently (which is repeated more times). If there is more than one, the sample is said to be bimodal (two fashions) or multimodal (more than two fashions). Any qualitative or quantitative variable can be calculated for any type of variable.

3. POSITION OR QUANTITY MEASURES

They are calculated for quantitative variables and, like the median, only take into account the ordered position of greater or lesser values ​​in the sample. The most common quantiles are percentiles, deciles, and quartiles, as they divide the ordered sample into 100 (perciles), 10 (deciles), or 4 parts (quartiles), respectively.

- Percentiles: Divide the ordered sample into 100 parts. The value of P50 corresponds to the median value.

- Deciles: Divide the ordered sample into 10 parts. The value of D5 corresponds to the value of the median and, therefore, the value of the P50.

- Quartile: Divide the ordered sample into 4 parts.

4. DISPERSION MEASURES

The information provided by the measures of central tendency is limited.

Example:

- Series 1: 18, 19, 20, 21, 22. (the first group is homogeneous because they have the closest ages)

- Medium series 1 = 20

- Series 2: 9, 14, 20, 27, 30.

- Medium series 2 = 20

What differentiates one from the other? The dispersion.

• Range or path: Difference between the highest and lowest value of the sample lXn-X1l (absolute value).

According to the previous example:

- R1 = 22-18 = 4

- R2 = 30-9 = 21 (this already indicates to us that the series 2 has more dispersion).

• Mean deviation: Arithmetic mean of the distances of each observation with respect to the sample mean:

• Standard or Standard Deviation: Quantifies the error we make if we represent a sample solely by its mean. This is the one that is most used because it gives us a greater range of error:

• Variance: Express the same information in quadratic values:

• Interquartile range: Difference between the third and the first quartile = lQ3-Q1l

- Coefficient of variation: It is a measure of relative dispersion (dimensionless). It serves us to compare the heterogeneity of two numerical series independently. It goes from 0 to 1.

5. NORMAL DISTRIBUTIONS

In statistics is called normal distribution, Gaussian distribution or Gaussian distribution, to one of the distributions of probability of continuous variable that more frequently appears in real phenomena. The normal distributions in a histogram show the Gaussian bell. And it is symmetrical with respect to the values ​​of central position, that is to say that the fashion will coincide with the mean and the median.

6. ASYMMETRY AND KURTOSIS

• Asymmetry coefficient of a variable: Asymmetry degree of the distribution of its data around its mean, the more asymmetric it is, the more different values ​​we will find.

Asymmetries:

The results can be as follows:

- g1 = 0 (symmetric distribution; there is the same concentration of values ​​to the right and left of the mean).

- g1> 0 (positive asymmetrical distribution, there is a higher concentration of values ​​to the right of the mean than to its left).

- g1 <0 (negative asymmetric distribution; there is a higher concentration of values ​​to the left of the mean than to the right).

• Kurtosis or pointing of the curve: It has no relation to symmetry. It serves to measure the degree of concentration of the values ​​it takes around its mean. The data accumulates a lot. A variable with normal distribution is chosen as reference, so that for it the kurtosis coefficient is 0.

The results can be as follows:

- g2 = 0 (mesocuric or normal distribution). It has a mean degree of concentration around the central values ​​of the variable (the same that has a normal distribution).

- g2> 0 (leptokurtic distribution). It presents a high degree of concentration around the central values ​​of the variable.

- g2 <0 (platicurtic distribution). It presents a reduced degree of concentration around the central values ​​of the variable.

7. TYPICATION OF VALUES AND THEIR RELATIONSHIP WITH THE GAUSS HOOD.

We work with a continuous variable that:

- Follows a normal distribution (TLC)

- It has more than 100 units (LGN)

The typing allows us to know if value corresponds or not to that distribution frequently

We know from the shape of the curve that:

The average matches the top of the bell: 8

The standard deviation is 2 points:
- 50% have scores> 8 - 50% have scores <8
- Approximately 68% score between 6 and 10