The problem presents data about the number of errors made by 7 candidates during a typing test. It asks to identify the population, statistical unit, observed character, and its nature. It also requires to represent the data graphically, calculate statistical measures like mode, median, quartiles, mean, variance, standard deviation, and coefficient of variation.

1. Problem Description

The problem presents data about the number of errors made by 7 candidates during a typing test. It asks to identify the population, statistical unit, observed character, and its nature. It also requires to represent the data graphically, calculate statistical measures like mode, median, quartiles, mean, variance, standard deviation, and coefficient of variation.

2. Solution Steps

1. Population, Statistical Unit, Character, and Nature:

* Population: The population studied is the set of 7 candidates who took the typing test.

* Statistical Unit: The statistical unit is each individual candidate.

* Observed Character: The observed character is the number of errors made by each candidate.

* Nature of the Character: The nature of the character is quantitative discrete since the number of errors can only take integer values.

2. Graphical Representation:

* Bar Diagram (Digramme en bâtons): A bar diagram represents the number of errors for each candidate. The x-axis represents the candidates (1 to 7), and the y-axis represents the number of errors. The height of each bar corresponds to the number of errors made by that candidate.

* Frequency Polygon (Polygone des fréquences): To construct a frequency polygon, we need to plot the number of errors against their respective frequencies. The data can be summarized as follows:

* 1 error: 1 candidate

* 3 errors: 1 candidate

* 4 errors: 1 candidate

* 5 errors: 1 candidate

* 6 errors: 1 candidate

* 7 errors: 1 candidate

* 10 errors: 1 candidate

Plot these points and connect them with straight lines.

* Cumulative Frequency Diagram (Diagramme cumulatif): The cumulative frequency is calculated by adding up the frequencies.

* 1 error or less: 1

* 3 errors or less: 2

* 4 errors or less: 3

* 5 errors or less: 4

* 6 errors or less: 5

* 7 errors or less: 6

* 10 errors or less: 7

Plot these cumulative frequencies against the corresponding number of errors. This can be represented as a step function or by connecting the points to show the cumulative distribution.

3. Mode, Median, Quartiles, Range, and Interquartile Range:

The data is: 1, 5, 4, 3, 7, 6,

1

0.

* Mode: The mode is the value that appears most frequently. Each value appears once in the dataset. Therefore, there's no mode (or all the data are the mode).

* Median: To find the median, first order the data: 1, 3, 4, 5, 6, 7,

1

0. The median is the middle value. Since there are 7 data points, the median is the (7+1)/2 = 4th value, which is

5.

* Quartiles:

* Q1: The first quartile is the median of the lower half of the data (excluding the median if the number of data points is odd). The lower half is 1, 3,

4. Q1 is the median of these three numbers, which is

3. * Q3: The third quartile is the median of the upper half of the data (excluding the median). The upper half is 6, 7,

1

0. Q3 is the median of these three numbers, which is

7.

* Range (Ecart): The range is the difference between the maximum and minimum values: 10 - 1 =

9.

* Interquartile Range (Intervalle Interquartile): The interquartile range is the difference between Q3 and Q1: 7 - 3 =

4.

4. Mean, Variance, Standard Deviation, and Coefficient of Variation:

* Mean (Moyenne Arithmétique): The mean is the sum of the values divided by the number of values:

$mean = (1 + 5 + 4 + 3 + 7 + 6 + 10) / 7 = 36 / 7 ≈ 5.14$

* Variance: The variance is the average of the squared differences from the mean:

$variance = \frac{\sum_{i=1}^{n} (x_i - mean)^2}{n}$

$variance = \frac{(1-5.14)^2 + (5-5.14)^2 + (4-5.14)^2 + (3-5.14)^2 + (7-5.14)^2 + (6-5.14)^2 + (10-5.14)^2}{7}$

$variance = \frac{17.1396 + 0.0196 + 1.2996 + 4.5796 + 3.4596 + 0.7396 + 23.6196}{7} = \frac{50.8572}{7} ≈ 7.265$

* Standard Deviation (Ecart-type): The standard deviation is the square root of the variance:

$standard\ deviation = \sqrt{variance} = \sqrt{7.265} ≈ 2.695$

* Coefficient of Variation: The coefficient of variation is the standard deviation divided by the mean, expressed as a percentage:

$CV = \frac{standard\ deviation}{mean} * 100 = \frac{2.695}{5.14} * 100 ≈ 52.43\%$

3. Final Answer

1. Population: The 7 candidates.

Statistical Unit: Each candidate.

Observed Character: Number of errors.

Nature of Character: Quantitative discrete.

2. Graphical representations: Described above.

3. Mode: None

Median: 5

Q1: 3

Q3: 7

Range: 9

Interquartile Range: 4

4. Mean: 5.14

Variance: 7.265

Standard Deviation: 2.695

Coefficient of Variation: 52.43%