Τα μαθηματικά είναι μια επιστήμη που εφαρμόζεται σε κάθε πτυχή της καθημερινής ζωής. Από την αμοιβή που λαμβάνουμε από την εργασία μας, το ενοίκιο και τους λογαριασμούς που πληρώνουμε, τις τιμές των καυσίμων και των προϊόντων που αγοράζουμε μέχρι τους τραπεζικούς λογαριασμούς και όλα στα στοιχεία της ΕΛΣΤΑΤ, τα πάντα εκφράζονται με αριθμούς.
Λένε όμως οι αριθμοί πάντα την αλήθεια;
Πριν απαντήσουμε στο παραπάνω ερώτημα θα πρέπει να κάνουμε έναν διαχωρισμό. Έχει σημασία λοιπόν εάν ο αριθμός που εξετάζουμε εκφράζεται ως απόλυτο νούμερο ή ως ποσοστό.
Έλα να κάνουμε μαζί μια άσκηση.
Έστω ότι ένας εργαζόμενος υφίσταται μείωση του μισθού του κατά 50% (ακραίο αλλά στην Ελλάδα μπορεί να συμβεί!). Έστω ότι μετά από λίγο καιρό τον φωνάζει ο εργοδότης του και του λέει ότι για να αποκαταστήσει την αδικία θα του αυξήσει το μισθό κατά 50%.
Ωραία σκέφτεται ο εργαζόμενος. Πριν λίγους μήνες έχασα το 50% του μισθού μου και τώρα το πήρα πίσω! Άρα αναμένει να λάβει το μισθό που λάμβανε αρχικά. Και δυστυχώς το ίδιο θα ανέμεναν και όλοι όσοι βρίσκονταν στην ίδια κατάσταση με αυτόν.
Στο τέλος του μήνα όμως, ο ήρωάς μας τσεκάρει τον τραπεζικό του λογαριασμό και βλέπει ότι του έχει κατατεθεί ως μισθός ένα ποσό μικρότερο από το αρχικό.
Τι συνέβη; Πού έκανε λάθος ο εργαζόμενος;
Πουθενά! Τα ποσοστά μείωσης και αύξησης όντως είναι ίδια, η βάση υπολογισμού όμως είναι διαφορετική.
Ας το ποσοτικοποιήσουμε για να το καταλάβουμε καλύτερα.
Έστω ότι ο φίλος μας λάμβανε αρχικό μισθό 1000€.
Με τη μείωση 50% που του έκανε ο εργοδότης ο μισθός του πλέον θα ήταν 50%*1000€=500€
Στη συνέχεια ο εργοδότης του έκανε αύξηση 50% άρα το μισθός του πλέον θα διαμορφωνόταν στα (1+50%)*500€=750€
Αν το παραπάνω παράδειγμα το βλέπαμε ως απόλυτα νούμερα και όχι ως ποσοστά θα λέγαμε ότι ο εργαζόμενος με τη μείωση έχασε 500€ και με την αύξηση κέρδισε 250€. Ενώ δηλαδή ως ποσοστά (μείωσης και αύξησης) είναι τα ίδια, μόνο όταν μετατραπούν σε απόλυτα ποσά φαίνεται η πραγματική διαφορά.
Για να επανέλθει δηλαδή ο εργαζόμενος στον αρχικό του μισθό θα πρέπει ο εργοδότης να του κάνει αύξηση 100% και όχι 50%, ακριβώς γιατί η βάση υπολογισμού είναι διαφορετική!
Κάτι αντίστοιχο συμβαίνει και με τα λεγόμενα ποσοστά κέρδους. Εάν έχουμε δηλαδή δύο εταιρείες που από τα έργα που έχουν αναλάβει να υλοποιήσουν κερδίζουν και οι δύο 20% τότε δημιουργείται η εντύπωση ότι τα κέρδη τους είναι τα ίδια. Μπορεί όμως η μία εταιρεία να κέρδισε 20% σε ένα έργο των 10.000€ ενώ η άλλη να κέρδισε 20% σε ένα έργο των 150.000€. Σε απόλυτα ποσά η πρώτη θα έχει κέρδη 2.000€ ενώ η δεύτερη 30.000€. Βέβαια είναι διαφορετικά και τα έξοδα που έχουν αλλά και πάλι η διαφορά σε απόλυτα νούμερα είναι τεράστια.
Συμπέρασμα: Όταν συγκρίνουμε μεταξύ τους ποσοστά θα πρέπει η βάση αναφοράς να είναι η ίδια.
Θέλεις να κάνουμε ακόμη μια άσκηση;
Έστω ότι έχουμε τους παρακάτω αριθμούς:
1, 4, 6, 10, 15, 18
Αν κάποιος θέλει να βρει τη μέση τιμή των παραπάνω αριθμών τότε θα υπολογίζει το γνωστό σε όλους μας μέσο όρο ως:
(1+4+6+10+15+18)/6=9
Τι θα γινόταν όμως εάν στην παραπάνω σειρά προσθέταμε στο τέλος και τον αριθμό 156;
Αν υπολογίζαμε και πάλι το μέσο όρο θα βρίσκαμε:
(1+4+6+10+15+18+156)/7=30
Βλέπουμε λοιπόν ότι στη δεύτερη περίπτωση ο μέσος όρος δεν είναι σωστό μέγεθος για να περιγράψει τη μέση τιμή ακριβώς επειδή υπάρχει μια ακραία τιμή στο δείγμα που στρεβλώνει το αποτέλεσμα.
Σε τέτοιες περιπτώσεις ή θα πρέπει να απομονώνουμε τις ακραίες τιμές και να υπολογίζουμε μετά το μέσο όρο, ή αντί για το μέσο όρο να υπολογίζουμε τη διάμεσο, δηλαδή τον αριθμό που βρίσκεται ακριβώς στη μέση εάν τοποθετήσουμε όλα τα νούμερα του δείγματος σε αύξουσα σειρά.
Στο παράδειγμά μας η διάμεσος είναι ο αριθμός 10 που είναι πολύ κοντά στο μέσο όρο χωρίς να λάβουμε υπόψη την ακραία τιμή (το νούμερο 9 δηλαδή).
Συμπέρασμα: Τελικά το μέγεθος μετράει!
Άρα τελικά οι αριθμοί λένε πάντα την αλήθεια;
Φυσικά είναι η απάντηση…αρκεί να ξέρεις να τους διαβάζεις!
Και ένα ακόμη παράδειγμα πριν κλείσουμε:
Έστω ότι το μήνα Απρίλιο μια χώρα είχε ρυθμούς ανάπτυξης 5%, το Μάιο 4% και τον Ιούνιο 3%.
Εάν βάλεις τα νούμερα σε ένα διάγραμμα θα προκύψει μια γραμμή που συνεχώς μειώνεται και έτσι δημιουργείται η εντύπωση ότι η χώρα έχει ύφεση.
Σωστό; Λάθος!
Ο μείωση του ρυθμού αύξησης ενός μεγέθους λοιπόν δεν σημαίνει ότι το μέγεθος μειώνεται αλλά ότι αυξάνεται πιο αργά.
Τέλος θα πρέπει να σημειώσουμε ότι όλοι οι παραπάνω μαθηματικοί υπολογισμοί χρησιμοποιούνται από την ανάποδη τόσο από τις κυβερνήσεις όσο και από τα ΜΜΕ (άλλοτε από αμέλεια και άλλοτε εσκεμμένα) ώστε να δημιουργούν εντυπώσεις που τις περισσότερες φορές μαθηματικά δεν στέκουν!
Εσύ τι γνώμη έχεις;
Πηγή εικόνας: https://pixabay.com/el/users/chenspec-7784448/
# Harvard University: A Legacy of Excellence and Innovation
## A Brief History of Harvard University
Founded in 1636, **Harvard University** is the
oldest and one of the most prestigious higher education institutions
in the United States. Located in Cambridge, Massachusetts,
Harvard has built a global reputation for academic excellence, groundbreaking research, and influential alumni.
From its humble beginnings as a small college established to educate clergy, it has evolved into a
world-leading university that shapes the future across various disciplines.
## Harvard’s Impact on Education and Research
Harvard is synonymous with **innovation and intellectual leadership**.
The university boasts:
– **12 degree-granting schools**, including the
renowned **Harvard Business School**, **Harvard Law School**, and **Harvard Medical School**.
– **A faculty of world-class scholars**, many of whom are Nobel laureates, Pulitzer Prize winners, and pioneers in their fields.
– **Cutting-edge research**, with Harvard leading initiatives in artificial
intelligence, public health, climate change, and more.
Harvard’s contribution to research is immense, with billions of dollars allocated to scientific discoveries
and technological advancements each year.
## Notable Alumni: The Leaders of Today and Tomorrow
Harvard has produced some of the **most influential figures** in history, spanning politics, business, entertainment, and science.
Among them are:
– **Barack Obama & John F. Kennedy** – Former U.S.
Presidents
– **Mark Zuckerberg & Bill Gates** – Tech visionaries (though Gates did not graduate)
– **Natalie Portman & Matt Damon** – Hollywood icons
– **Malala Yousafzai** – Nobel Prize-winning activist
The university continues to cultivate future leaders who shape industries and drive global progress.
## Harvard’s Stunning Campus and Iconic Library
Harvard’s campus is a blend of **historical charm and modern innovation**.
With over **200 buildings**, it features:
– The **Harvard Yard**, home to the iconic **John Harvard Statue** (and the famous “three lies” legend).
– The **Widener Library**, one of the largest university libraries in the world,
housing **over 20 million volumes**.
– State-of-the-art research centers, museums, and performing arts venues.
## Harvard Traditions and Student Life
Harvard offers a **rich student experience**, blending academics with vibrant traditions, including:
– **Housing system:** Students live in one of 12 residential
houses, fostering a strong sense of community.
– **Annual Primal Scream:** A unique tradition where students de-stress
by running through Harvard Yard before finals!
– **The Harvard-Yale Game:** A historic football rivalry
that unites alumni and students.
With over **450 student organizations**, Harvard students engage in a diverse range of
extracurricular activities, from entrepreneurship to performing arts.
## Harvard’s Global Influence
Beyond academics, Harvard drives change in **global policy,
economics, and technology**. The university’s research impacts healthcare, sustainability, and artificial intelligence,
with partnerships across industries worldwide. **Harvard’s endowment**, the largest of any university,
allows it to fund scholarships, research, and public initiatives, ensuring a legacy of impact
for generations.
## Conclusion
Harvard University is more than just a school—it’s a **symbol of
excellence, innovation, and leadership**. Its **centuries-old traditions, groundbreaking discoveries, and transformative education** make it one of the most influential institutions in the world.
Whether through its distinguished alumni, pioneering research, or vibrant student
life, Harvard continues to shape the future in profound ways.
Would you like to join the ranks of Harvard’s legendary scholars?
The journey starts with a dream—and an application!
https://www.harvard.edu/
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