2.7.1 · D5 · HinglishStatistics & Probability — Intermediate
Question bank — Measures of central tendency — mean (grouped - ungrouped), median (grouped), mode (grouped)
2.7.1 · D5· Maths › Statistics & Probability — Intermediate › Measures of central tendency — mean (grouped - ungrouped), m
True or false — justify
The mean is always one of the actual data values.
False. Mean ek balance point hai, data ka member nahi — jaise ka mean hai, jo kabhi occur nahi hota. Ye values ke beech mein rehta hai.
The median must be one of the actual data values.
False for even (ye do middle values ka average leta hai, jaise ) aur grouped data ke liye bhi false hai (interpolation median class ke andar kahin bhi land kar sakti hai). Ye member tabhi hota hai jab odd ho aur data ungrouped ho.
Adding a huge outlier changes the median a lot.
False. Median sirf sorted list mein position ki parwah karta hai, isliye ek extreme value middle ko barely shift karti hai; mean hi outlier ki taraf khicha jaata hai.
If all frequencies are equal, the grouped mode formula gives the modal class centre.
False — equal frequencies hone par koi single peak nahi hoti; aur zero ho sakte hain, jo formula ko undefined ya meaningless bana deta hai. Distribution ka koi mode nahi hota.
For a perfectly symmetric distribution, mean, median and mode coincide.
True. Symmetry ka matlab hai ki balance point, middle, aur peak sab symmetry ke axis par baith jaate hain, isliye teeno ek hi value par land karte hain.
The empirical relation is an exact identity.
False. Ye sirf moderately skewed unimodal data ke liye valid ek approximation hai; strongly skewed ya multi-modal data ke liye ye kaafi off ho sakta hai.
Doubling every observation doubles the mean, median and mode.
True. Teeno linear location measures hain, isliye har value ko constant se scale karne par har measure bhi se scale hota hai (ye shifted/stretched picture ko track karte hain).
Adding 5 to every observation adds 5 to the mean but leaves the median unchanged.
False. Ek constant add karne se sab kuch 5 se right ki taraf shift ho jaata hai, isliye mean, median aur mode teeno 5 se increase hote hain.
The class with the largest cumulative frequency is the modal class.
False. Cumulative frequency hamesha badhti rehti hai, isliye uski sabse badi value hamesha last class ki hogi. Modal class sabse badi individual frequency use karta hai.
Spot the error
A student writes the median class as "the class containing the -th item". Fix it.
rule ungrouped median position ke liye hai. Grouped data ke liye hum use karte hain (continuous interpolation), pehli class dhundhte hain jiska cumulative frequency ho.
Classes are and a student takes for the second class. What's wrong?
Ye inclusive limits hain jisme 29 aur 30 ke beech ek gap hai. Pehle continuous boundaries mein convert karna zaroori hai (), isliye true lower boundary hai, na ki .
In the mode formula a student uses in the denominator too, writing . Why is that wrong?
Denominator mein total excess dono neighbours se hona chahiye, . Sirf ek side use karne se peak ke baad wali class ka pull ignore ho jaata hai.
To find the median, a student takes the middle of the list as . Spot the mistake.
List sort nahi ki gayi thi. Pehle order karo: , phir middle do values aur hain, jisse median milta hai — nahi.
A student computes grouped mean using class limits (20, 30) instead of midpoints. Why does this bias the answer?
Mean assume karta hai ki class ki har value uske centre par baithe. Limit use karne se har class ki assumed value systematically ek edge par shift ho jaati hai, jo poore mean ko zyada neeche (ya zyada upar) kheench deta hai.
Someone claims the step-deviation method gives a different mean than the direct method. Correct them.
Ye bilkul same mean deta hai. Step-deviation sirf algebra hai () jo arithmetic easy karne ke liye rearrange kiya gaya hai — koi information lose ya approximate nahi hoti.
Why questions
Why does the grouped median formula use as the "width per item"?
Hum assume karte hain ki median class ke items width mein evenly spread hain, isliye har item width occupy karta hai; isko us kitne items se multiply karte hain jo hume boundary ke baad chahiye, ye batata hai ki kitna andar step karna hai.
Why can a mean be non-representative of a skewed data set?
Kuch extreme values balance point ko long tail ki taraf kheench leti hain, isliye "fair share" ab ek typical observation ko describe nahi karti — isliye skewed data ke liye median ya mode prefer kiya jaata hai. Dekho Skewness.
Why do we assume every value in a class equals its midpoint for the grouped mean?
Individual values kho chuke hain, isliye midpoint sabse kam-biased guess hai: average par, centre ke upar aur neeche ki values cancel ho jaati hain, jo estimate ko roughly symmetric classes ke liye unbiased rakhta hai.
Why does the mode formula lean the peak left when the previous class is heavier ()?
Pehle ek heavier class ka matlab hai ki peak se bilkul pehle zyada "mass" hai, isliye underlying shape ka smooth peak modal class ki left edge ke zyada paas baith ta hai — fraction zero ki taraf shrink ho jaata hai.
Why is the median found from an ogive at the height ?
Ogive cumulative frequency plot karta hai; total ke aadhe par cross karne se wo value milti hai jahan exactly aadha data neeche hai — ye bilkul median ki definition hai. Dekho Cumulative Frequency & Ogives.
Why does setting define the balance point?
Ye kehta hai ki ke upar wali values ka total "pull" ke neeche wale pull ko exactly cancel karta hai, jo ek see-saw ke par balance karne ki physical condition hai — isse solve karne par milta hai.
Why do all three measures agree for a symmetric distribution but split apart under skew?
Symmetry peak, middle aur balance point ko ek axis par force karta hai; skew ek tail ko stretch karta hai, mean ko sabse zyada move karta hai (wo tail ko feel karta hai), mode ko sabse kam (wo sirf crowd dekhta hai), aur median beech mein rehta hai. Dekho Skewness.
Edge cases
What is the mode of the data where every value appears once?
Koi mode nahi hai — koi bhi value doosre se zyada crowded nahi hai. Ek data set ka zero, ek, ya kai modes ho sakte hain.
If a data set is , how many modes are there?
Teen modes () — ye multimodal hai. Single-peak grouped mode formula yahan invalid hai kyunki koi unique modal class nahi hai.
The modal class is the first class in the table, so there is no "class before". What do you do?
Missing ko maano. Formula phir bhi kaam karta hai: peak simply lower boundary se aage left nahi ja sakta.
The median falls exactly on a class boundary (i.e. before equals ). What does the formula give?
Numerator ho jaata hai, isliye Median exactly milta hai — beech wala person lower boundary par hi baith ta hai, jo formula saaf taur par respect karta hai.
For a single data point , what are the mean, median and mode?
Teeno ke barabar hain. ke saath balance point, middle, aur akeli value trivially coincide karti hain — kisi bhi formula ke liye ek useful sanity check.
What happens to the mode formula if (flat top)?
Denominator ho jaata hai, isliye formula undefined hai — koi rise nahi hai jisse peak locate ki ja sake, jo is fact se match karta hai ki flat region ka koi single mode nahi hota.
If two classes tie for the highest frequency, which is the modal class?
Koi bhi uniquely nahi — distribution bimodal hai aur single-peak formula apply nahi karna chahiye. Dono peaks report karo ya tie break karne ke liye finer grouping use karo.
Recall Quick self-test
The mean is dragged by outliers but the median isn't — why? ::: Mean values sum karta hai (ek outlier apna poora magnitude contribute karta hai), jabki median positions count karta hai (outlier bas "list ke end mein ek aur item" hai). A distribution with no repeated values has what mode? ::: Koi mode nahi — koi bhi value kisi doosre se zyada frequent nahi hai. Whose lower boundary do the median AND mode formulas need — limit or boundary? ::: Continuous boundary, kabhi bhi printed inclusive limit nahi.