U11. Statistics

Situación introductoria

Haz la siguiente encuesta para tener algunos datos reales con los que trabajar

No pulses "Intro", ni el botón "Enviar", hasta que completes el formulario del todo.

Saberes básicos

1. Variables estadísticas / statistical variables

2. Recuentos / tallies

3. Tablas estadísticas / statistical tables

4. Parámetros estadísticos / statistical parameters

   1. Proporcion / proportion

   2. Media, mediana, moda / mean, median, mode

   3. Desviación típica / standard deviation

   4. Rango / range

5. Gráficas estadísticas / statistical charts

   1. Diagrama de barras / bar chart

   2. Polígono de frecuencias / line chart

   3. Diagrama de sectores / pie chart

Statistical variables

Statistical variables: characteristics or properties that can take on different values and are used to collect and analyze data.

• • Qualitative variables: Variables that describe categories or qualities, not measured with numbers (e.g., color, brand, favorite subject).

  • Quantitative variables: Variables that represent numerical values and can be measured or counted (e.g., height, number of siblings).

    • Discrete variables: Quantitative variables that take specific, separate values (usually whole numbers), like the number of pets or siblings.

    • Continuous variables: Quantitative variables that can take any value within a range, often involving measurements (e.g., weight, height, time).

Exercise

Classify the statistical variables from the survey you filled at the beginning of the unit. Sort them into qualitative and quantitative (discrete and continuous) variables:

white half-moons on nails, siblings, math notebook color, weight, height, painted nails, left-handed or right-handed, video game console, favorite subject, worst subject, how often you eat fish, times of video games, Instagram followers, mobile screen state, recycling, football team, phone operative system, shoe size, heart rate, household chores, pieces of fruit, hours doing sport.

Continuous variables

Activity

Create a statistical table, using a spreadsheet, for each of the following data distributions collected from the survey completed at the beginning of the unit (3rd ESO, school year 2024–2025):

• How many brothers and sisters do you have?: 1, 4, 2, 2, 2, 2, 2, 3, 2, 2, 2, 1, 3, 4, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2

• What colour are Mathematics for you?: Red, Red, Blue, Red, Blue, Red, DK/NA, Red, Red, Red, Red, Blue, Blue, Blue, Blue, Red, Red, Blue, Blue, Red, Red, Red, Red, Red, Blue

• Are you left-handed or right-handed?: Right-handed, Left-handed, Right-handed, Ambidextrous, Right-handed, Right-handed, Right-handed, Right-handed, Right-handed, Right-handed, Right-handed, Right-handed, Right-handed, Left-handed, Right-handed, Right-handed, Right-handed, Right-handed, Right-handed, Right-handed, Right-handed, Right-handed, Right-handed, Right-handed, Right-handed

• Do you have white half-moons on your nails?: Yes, Yes, Yes, DK/NA, Yes, No, No, Yes, Yes, Yes, Yes, Yes, No, Yes, Yes, No, Yes, Yes, No, Yes, No, No, No, Yes, No

• How much do you weigh?: DK/NA, 59, 64, 65, DK/NA, 50, 62, 61, 48, 59, 50, 50, 54, 49, DK/NA, 70, 80, 54, 56, 54, 47, DK/NA, DK/NA, 50, 49

• How many times a week do you play videogames?: 0, 5, 3, 7, 1, 0, 3, 0, On the weekend, 2, 0, 0, 3, 0, 0, 4, 7, 7, 7, 0, 0, 0, 0, 3, DK/NA

• Have you got any video game console?: Sony (PS2, PS3, PS4, PS5), Wii, Nintendo Switch, Sony (PS2, PS3, PS4, PS5), Sony (PS2, PS3, PS4, PS5), Sony (PS2, PS3, PS4, PS5), None, Sony (PS2, PS3, PS4, PS5), Wii, Nintendo Switch, Xbox, Other, Wii, None, Sony (PS2, PS3, PS4, PS5), Sony (PS2, PS3, PS4, PS5), Sony (PS2, PS3, PS4, PS5), Sony (PS2, PS3, PS4, PS5), Wii, Sony (PS2, PS3, PS4, PS5), None, None, Sony (PS2, PS3, PS4, PS5), Wii

• Do you have your nails painted?: No, No, No, No, No, Sometimes, No, No, Sometimes, No, No, Sometimes, No, Sometimes, No, Yes, No, Sometimes, No, No, No, Sometimes, No, No, No, No, No

• What are your favourite subjects? (pick as many as you want): Biology Lab, English Language, Mathematics, Technology and Digitization, Biology and Geology, Biology Lab, English Language, Technology and Digitization, Biology and Geology, Biology Lab, Geography and History, Phisical Education, Geography and History, Mathematics, Phisical Education, Technology and Digitization, Flamenco Culture, Phisical Education, Geography and History, Mathematics, Spanish Language, Biology and Geology, Philosophy and Debate, Spanish Language, Arts, Biology and Geology, English Language, Geography and History, Spanish Language, Technology and Digitization, Biology Lab, Mathematics, Phisical Education, Biology and Geology, Flamenco Culture, Biology and Geology, Classic Culture, Biology Lab, Geography and History, Philosophy and Debate, Arts, Biology Lab, Technology and Digitization, English Language, Flamenco Culture, Phisical Education, Biology and Geology, English Language, Flamenco Culture, Biology and Geology, Biology Lab, Computer Studies and Robotics, English Language, Geography and History, Mathematics, Philosophy and Debate, Technology and Digitization, Arts, Computer Studies and Robotics, Geography and History, Phisical Education, Technology and Digitization, Arts, English Language, Flamenco Culture, Geography and History, Spanish Language, Flamenco Culture, Spanish Language, Biology and Geology, Biology Lab, Mathematics, Technology and Digitization, Arts.

• What is your height?: DK/NA, 170, 170, 170, 120, 148, 164, 166, 174, 170, 160, 156, 166, 178, 155, 177, 174, 171, 169, 165, 160, 156, 160, 165, 158.

• How many times a week do you eat fish?: 1, 1, 2, 1.5, 1, 1, 2, 1.5, 1, 1,3, 3, 5, 2, 1, 1, 2, 1, 2, 2, 1, 0.5, 1, 0

• How many hours a day do you watch TV?: 0.5, 1, 1, 0, 2, 1, 0.5, 1, 0.6, 0, 1, 1, 0.5, 1, 0, 0, 0, 0, 0.5, 0.5, 0, 0, 0.5

• How many hours a day do you watch your smartphone?: 1, 3, 2.3, 4, 57908, 0, 14, 6, 1, 7, 2, 4, 6, 4, 2, 5, 3, 3, 4, 12, 12, 2.5, 3, 3

• What are your worst subjects? (pick as many as you want): Arts, Mathematics, Phisics and Chemistry, Arts, Arts, Flamenco Culture, Phisics and Chemistry, Arts, Biology and Geology, Biology Lab, Classic Culture, Computer Studies and Robotics, English Language, Flamenco Culture, Geography and History, Mathematics, Philosophy and Debate, Phisical Education, Phisics and Chemistry, Spanish Language, Technology and Digitization, Geography and History, Phisics and Chemistry, Spanish Language, Arts, English Language, English Language, Mathematics, Phisics and Chemistry, Mathematics, Phisics and Chemistry, English Language, Spanish Language, English Language, Mathematics, Phisics and Chemistry, Geography and History, Mathematics, Phisics and Chemistry, Mathematics, Phisics and Chemistry, Mathematics, Phisics and Chemistry, English Language, Mathematics, Phisics and Chemistry, Mathematics, Phisics and Chemistry, Arts, Classic Culture, Flamenco Culture, Phisical Education, Phisics and Chemistry, Spanish Language, Biology and Geology, English Language, Mathematics, Phisics and Chemistry, Spanish Language, Biology and Geology, Biology Lab, Classic Culture, Computer Studies and Robotics, English Language, Flamenco Culture, Geography and History, Mathematics, Philosophy and Debate, Phisical Education, Phisics and Chemistry, Spanish Language, Technology and Digitization, Biology and Geology, Mathematics, Phisics and Chemistry, Biology and Geology, English Language, Arts, Computer Studies and Robotics, English Language, Geography and History, Phisical Education, Technology and Digitization, Biology and Geology, Biology Lab, Phisics and Chemistry, Biology and Geology, Mathematics, Phisics and Chemistry.

• How many followers do you have in Instagram: 500, 66, 250, 450, 1500, 800, 157, 348, 37, 650, 1163, 450, 760, 78, 0, 300, 300, 145, 82, 30, 666, 570, 500, 700, 67

• And how many people do you follow?: 2500, 120, 500, 150, 200, 500, 800, 38, 69, 1100, 198, 600, 350, 128, 0, 300, 500, 3, 115, 15, 536, 660, 1250, 89, 100

• What would you like to be when you grow up?: A millionaire, rich, teacher of Biology or Physical Education, millionaire, psychologist, journalist, millionaire, lawyer, a teacher of Spanish Language (Primary School), civil engineer or similar, I don't know, psychologist, millionaire, History teacher, teacher, businessman, marketing team, cruiser captain, Formula 1 pilot, administrator, I would like to be an English teacher in high school, I would like to be a primary teacher or journalist, nursing, millionaire, I don't know.

• How is your phone screen?: Intact, Intact, Broken, Broken, Intact, Intact, Intact, Intact, Broken, Intact, Broken, Intact, Broken, Broken, Intact, Broken, Intact, Broken, Intact, Broken, Intact, Intact, Intact, Intact, Intact.

• Which wastes do you recycle at home?: None, Paper, Plastics, Used oil, Plastics, Batteries, Medicines, Plastics, Light bulbs, Batteries, Used oil, Light bulbs, None, Batteries, Used oil, Light bulbs, Paper, Plastics, Used oil, Glass, Light bulbs, Medicines, Plastics, Used oil, Paper, Used oil, Paper, Plastics, Glass, Paper, Glass, Used oil, Glass, Used oil, Paper, Plastics, Glass, Plastics, Light bulbs, Paper, Plastics, Glass, Batteries, Light bulbs, Paper, Plastics, Glass, Batteries, Used oil, Light bulbs, Paper, Plastics, Glass, Batteries, Used oil, None, None, Plastics, Glass, Batteries, Used oil, Light bulbs, Medicines, None, Paper, Plastics.

• Which is your favourite football team?: Granada C. F., Granada C. F., Real Madrid, Granada C. F., Granada C. F., Atlético de Madrid, Granada C. F., Real Madrid, Real Madrid, F. C. Barcelona, Granada C. F., Real Madrid, Granada C. F., Real Madrid, Granada C. F., Real Madrid, Granada C. F., Granada C. F., Granada C. F., F. C. Barcelona, I don't have any, F. C. Barcelona, F. C. Barcelona, Otro, Granada C. F., Atlético de Madrid, F. C. Barcelona, Real Madrid, Granada C. F., Real Madrid, Granada C. F., Real Madrid, Granada C. F., F. C. Barcelona, Granada C. F., Real Madrid, Real Madrid, Granada C. F.

• How much money do you carry a normal working day?: 0, 2, 10, I don't know, depends, 0, 10, 5, 2, 10, 5, 1, 25, 1, 3, 0, 10, 0, 40, 0, 30, 40, 0, 30, 0

• What is your phone operative system?: Android, Android, iOS, iOS, iOS, Android, Android, iOS, Android, iOS, iOS, iOS, Android, Android, iOS, Android, Android, Android, Android, Android, iOS, iOS, Android, iOS, Other

• How many devices connected to the internet are there at home?: 4, 11, 10, 6, 12, 8, 15, 7, 13, 11, 3, 10, 4, 7, 6, 5, 9, 18, 15, 15, 7, 6, 12, 13, 9

• Which is your shoe size?: 43, 43, 40, 45, 37, 36, 41, 41, 38, 41, 37, 38, 40, 42, 38, 43, 42, 42, 45, 43, 38, 38, 39, 38, 36

• Measure your heart rate (pulses/min.): 88, 52, 60, 60, 120, 100, 72, 88, 68, 98, DK/NA, 130, 32, 84, 28, 76, 76, 56, 60, DK/NA, 64, DK/NA, 40, 70, 42

• Which kind of household chores do you do at home?: Sweep, vacuum or scrub, Take care of my little brothers or sisters, Make the bed, Load or unload the dishwasher, Set or clear the table, Take care of my little brothers or sisters, Cook, Set or clear the table, Others, Make the bed, Cook, Load or unload the dishwasher, Make the bed, Cook, Set or clear the table, Make the bed, Make the bed, Do the shopping, Set or clear the table, Sweep, vacuum or scrub, Make the bed, Do the shopping, Cook, Set or clear the table, Sweep, vacuum or scrub, Make the bed, Cook, Do the laundry or hang out, Set or clear the table, Sweep, vacuum or scrub, Take care of my little brothers or sisters, Make the bed, Load or unload the dishwasher, Set or clear the table, Others, Sweep, vacuum or scrub, Make the bed, Load or unload the dishwasher, Set or clear the table, Make the bed, Do the laundry or hang out, Make the bed, Cook, Do the laundry or hang out, Others, Sweep, vacuum or scrub, Make the bed, Cook, Load or unload the dishwasher, Set or clear the table, Others, Do the shopping, Cook, Load or unload the dishwasher, Set or clear the table, Make the bed, Do the shopping, Cook, Do the laundry or hang out, Set or clear the table, Sweep, vacuum or scrub, Take care of my little brothers or sisters, Do the shopping, Cook, Load or unload the dishwasher, Others, Take care of my little brothers or sisters, Make the bed, Do the shopping, Do the laundry or hang out, Load or unload the dishwasher, Set or clear the table, Sweep, vacuum or scrub, Take care of my little brothers or sisters, Make the bed, Do the shopping, Cook, Do the laundry or hang out, Load or unload the dishwasher, Set or clear the table, Others, Sweep, vacuum or scrub, Make the bed, Do the shopping, Cook, Load or unload the dishwasher, Set or clear the table, Make the bed, Do the shopping, Others, Make the bed, Cook, Make the bed, Cook, Make the bed, Set or clear the table, Others, None of them, Make the bed, Set or clear the tablethe bed, Cook, Do the laundry or hang out, Others

• How many pieces of fruit do you eat per week?: 4, 12, 3, 5 or 6, dpend, 3, 5 o 6, one every day, 5 or 6, 7, 4, Three, 3, 8, 2o3, 7, 7, 1, 8, 15, twice, 4 or 5, 7, 0-2, 6

• How many hours per week do you play any sport?: 4.5, 1, 2, 4 or 5, tumbing, 2.30, 11.30, 9, Any :), 5, 9, None, 3, 0, 0, 8, 4, 7, 8, 10, i don´t play any sport, I don`t play any sport, 4, 1.30h, 4.

• How many gums are stuck under your desk?: 4, 6, 1, 1, 2, 2, 2, 1, 9, 0, 7, Nothing, 0, 0, 2, 8, none, 11, 4, 5, 6, 3, 5, 3, 0

Parameters

• Measures of central tendency: mean, weighted mean, median, mode, percentiles.

• Measures of dispersion: range, variance, standard deviation, coefficient of variation, box plot.

Mode

The mode is the value that appear most often in a set of data.

The mode is the value of the variable that occurs most frequently, the one that repeats the most, the one with the highest absolute frequency. It is the simplest parameter to calculate and can be determined for all types of variables, even those expressed with words instead of numbers, called qualitative variables.

It is closely related to the proportions or percentages we calculated in the section on statistical tables, in the sense that it is the value with the highest percentage, the largest proportion.

Arithmetic mean / average

The mean is a value of the variable that marks its balance point. It summarizes all the collected values into a single one.

The arithmetic mean allows easy comparison between two populations. For example, students in 4th grade and those in 3rd grade; 4th-grade students at Fray Luis and those at another school; or 4th-grade students this year and those from last year.

Its calculation can only be done for quantitative variables. It is calculated by adding all the values and dividing by the total number of collected values.

Pay attention to this number: teenagers eat an average of 5.3 pieces of fruit per week.

Maybe someone can eat one whole piece of fruit and part of another, but an arithmetic mean does not represent a specific individual value.

Activity: find out the arithmetic mean of the quantitative variables in our study. 

The red line shows the median height of this group of dolls.

For a series of numbers, finding the median is simple. First, you order them, then identify the one in the middle.

For example, the number of connected devices at home in our class is, according to the initial survey, are: 

4, 11, 10, 6, 12, 8, 15, 7, 13, 11, 3, 10, 4, 7, 6, 5, 9, 18, 15, 15, 7, 6, 12, 13, 9

These 25 numbers, ordered from smallest to largest, are:

3, 4, 4, 5, 6, 6, 6, 7, 7, 7, 7, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 15, 15, 15, 18

and the number in the middle is 9.

That value, precisely, is the median. Its interpretation is as follows: someone with 9 devices has as many classmates with more connected devices at home as with the same or fewer.

When the data is arranged in an ordered table, it's as if we already have them counted, so all we need to do is accumulate the frequencies of each variable value until we surpass half of the total values.

Activity:

Calculate the median heart rate of your classmates based on these collected data: 88, 52, 60, 60, 120, 100, 72, 88, 68, 98, DK/NA, 130, 32, 84, 28, 76, 76, 56, 60, DK/NA, 64, DK/NA, 40, 70, 42

The following video summarizes how to calculate the mean, the median, and the mode:

Actividades sobre las medidas de centralización / activities about central tendency measurements

1. Revisa el vídeo de la introducción, detecta la media, la mediana y la moda en todas las ocasiones en las que aparece.

2. Imagina que tu amigo se ha comido dos chuches y tú ninguna. Calcula la media de chuches que os habéis comido entre ambos. ¿Es representativo ese dato para ti?

3. Find measures of central tendency in this site.

4. Calculate the average of tweets per week of three of the people of this list.

5. The following numbers represents the time in minutes to get to school of some students (actually they are data from 1ESO students, year 12-13): 15, 20, 25, 15, 20, 15, 17, 17, 10, 15, 15, 30, 60, 30, 15, 15, 45, 23, 30, 20, 35, 30, 35, 35, 15, 15, 25, 15, 10, 10, 30, 20, 30, 5, 15, 15, 10, 20, 15, 15, 40, 10, 15, 30, 20, 10, 30, 20, 30, 45, 55, 60, 10, 15, 15, 20, 15, 55, 76, 30, 20.  Do you think they live far or close to the school?

6. Compara el peso medio de este curso con el obtenido en el curso anterior. ¿Obtienes alguna conclusión? 

7. Observa las siguientes estaturas de una muestra aleatoria del alumnado de 1º de ESO de este centro (curso 2012-2013): 154, 157, 165, 162, 162, 155, 155, 167, 157, 159, 152, 168, 168, 170, 162, 168, 160, 175, 150, 145, 140, 155, 155, 167, 155, 140, 140, 150. Los resultados para 2º de ESO (2017) han sido los siguientes: 162 155 157 168 1'75 m 170 162 cm 158 156 160 165 1,6 160 cm 1m 60cm 168 150 162 166 167 158 164 1,62 167 170cm 170 1,75-77 176 154 170 1.62 1.68m 165 167 cm. 1'60 1.67 148 1.62 170 153 1,55 150 178 178 160 1,69 160 158 161 1,55 171 1'60 171 171 170 1.65. Responde de un modo científico a esta pregunta ¿son más altos los chicos y chicas de 2º de ESO que los de 1º de ESO? ¿Qué pasa si los comparas con los datos de 4º que aparecen más arriba?

8. Haz diez intentos de estimación de 10 segundos sin ninguna práctica previa. Practica durante un rato y realiza otros diez intentos. Pasa todos los datos a una hoja de cálculo. ¿Crees que el entrenamiento ha mejorado tu capacidad de estimación? 

9. Razona la siguiente afirmación "la mayoría de habitantes españoles tiene más piernas que la media".

10. Observa la siguiente noticia (clic en la imagen para verla más grande), especialmente las partes resaltadas. ¿Hay algún error en el cálculo de la media?

Gráficos estadísticos

• Diagrama de sectores.

• Diagrama de barras.

• Polígono de frecuencias.

• Histograma (marcas de clase).

Abajo puede verse, un diagrama de sectores que muestra el color de las matemáticas para el alumnado encuestado. Véase sinestesia, aunque el diagrama corresponde más bien a gustos o manías.

Ejemplo con variable cuantitativa agrupada en intervalos: seguidores en Instagram

A partir de aquí, la página está pendiente de actualización

Nota: Esta unidad se imparte con la metodología del aprendizaje basado en proyectos según la planificación expresada aquí. 

La descripción del proyecto está en esta página.

Situación introductoria

Mira los siguientes vídeos

Recursos necesarios

Nota: Para este tema necesitarás los siguientes programas o apps (son sólo recomendaciones, cualquier programa con funciones similares es también válido):

• Una hoja de cálculo (spreadsheet), esto es, un programa que nos ayudará con los tediosos cálculos que exige la estadística. Hay varias posibilidades para ordenador y móvil. A continuación se enlazan algunos programas de software libre y gratuitos:

  • Hojas de Cálculo de Google (recomendada), Libreoffice y Openoffice (gratis, para ordenador), Excel (de Microsoft, la más popular, pero también más cara).

  • Hojas de Calculo de Google (recomendada) y Documents Free para Android e iOS. 

  • Otras posibilidades que funcionan online, sin registro o descargar nada o con un registro mínimo son zoho, ethercalc u offidocs.

• Un programa para realizar encuestas o cuestionarios: 

  • Google Forms.

  • Survey monkey.

• Otros programas para hacer encuestas en vivo son:

  • Direct poll.

  • Encuestas de twitter.

  • Kahoot surveys.

  • Quizizz surveys.

• Procesadores de textos (elige el que prefieras):

  • Libreoffice writer, Microsoft Word, Google docs...

• Programa para hacer presentaciones:

  • Presentaciones de Google.

  • Prezi.

  • Powerpoint (de Microsoft, no es gratis).

Enlaces

• Información sobre percentiles de talla y peso en España.

• Lectura recomendada: ¡A la moda! Y a la mediana, y a la media...

• La esperanza de vida, una media muy especial.

• ¿Es tu casa grande o pequeña?

• Estadística nombres españoles.

• Con medias y a lo loco.

• Sección de estadísticas de la Consejería de Educación de la Junta de Andalucía.

• Software: hoja de calculo.

• Save the pies for the desserts.

• Survey data and answers summary.

• World's median ages.

• Mistery boxes.

• Juega con las estadísticas: 94% android game.

• 38 facts about the Internet.

• Sobre los zurdos.

• Brecha salarial en España.

• Sobre el salario medio y el mediano.

• Cryptography.

• Media, mediana y moda en los salarios.

Preguntas del examen

El examen consistirá en hacer un análisis estadístico en una situación real pasando por todas las etapas del mismo: realizar un recuento, construir una tabla con todas las columnas, calcular los parámetros estadísticos y realizar un gráfico.

Vocabulary

1. Variable estadística – Statistical variable

2. Cualitativa – Qualitative

3. Cuantitativa – Quantitative

4. Discreta – Discrete

5. Continua – Continuous

6. Población – Population

7. Muestra – Sample

8. Frecuencia – Frequency

9. Frecuencia absoluta – Absolute frequency

10. Frecuencia relativa – Relative frequency

11. Frecuencia acumulada – Cumulative frequency

12. Tabla de frecuencias – Frequency table

13. Diagrama de barras – Bar chart

14. Diagrama de sectores – Pie chart

15. Histograma – Histogram

16. Media – Mean (o average)

17. Mediana – Median

18. Moda – Mode

19. Rango – Range

20. Intervalo – Interval

21. Marca de clase – Class mark

22. Distribución – Distribution

23. Datos – Data

24. Encuesta – Survey

25. Gráfico – Graph / Chart

26. Porcentaje – Percentage

27. Probabilidad – Probability