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Statistics

Practice MCQs

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Quantitative Aptitude Statistics Competitive Exams

Statistics is an important quantitative aptitude topic based on collection, organization, presentation, analysis, and interpretation of data. It includes concepts such as mean, median, mode, range, frequency distribution, variance, standard deviation, and graphical representation of data. These questions are commonly asked in competitive exams, entrance tests, banking exams, SSC, railway exams, and data interpretation sections.

What is Statistics?

Statistics is the study of data. It helps us collect numerical facts, arrange them properly, summarize them, and draw useful conclusions.

In aptitude exams, statistics questions usually test your ability to calculate averages, find the middle value, identify the most repeated value, compare spread of data, and read tables or charts correctly.

Quick idea: Mean gives the average, median gives the middle value, and mode gives the most repeated value.
Concept Meaning Example
Data Collection of facts or numbers Marks of students
Mean Average of all observations \( \frac{10+20+30}{3}=20 \)
Median Middle value after arranging data Median of 3, 5, 7 is 5
Mode Most repeated observation Mode of 2, 3, 3, 5 is 3
Range Difference between highest and lowest value Range of 4, 8, 10 is 6

“In statistics, first arrange the data properly. Correct arrangement avoids most mistakes.”

Aptitude Tip
Key points
  • Statistics deals with data.
  • Mean is the arithmetic average.
  • Median is the middle value.
  • Mode is the most frequent value.
  • Range measures spread.
  • Charts and tables help in quick interpretation.
data mean median mode range

Visual Understanding

These diagrams show how data can be summarized using mean, median, mode, and range.

Mean as Average
10 20 30 = 20 Mean balances the data into an average value
\[ \text{Mean}=\frac{\text{Sum of observations}}{\text{Number of observations}} \]

Mean gives a single representative value for the whole data set.

Median as Middle Value
Arrange data in ascending order 3 5 7 9 11 Median = 7
\[ \text{Median}=\text{Middle observation after arranging data} \]

Median divides the arranged data into two equal parts.

Mode as Most Repeated Value
Data: 2, 4, 4, 4, 6, 8 2 4 4 4 6 8 4 appears most often, so Mode = 4
\[ \text{Mode}=\text{Most frequent observation} \]

Mode is useful when we want to know the most common value.

Range as Spread
4 6 9 12 Range = 12 - 4 = 8
\[ \text{Range}=\text{Highest value}-\text{Lowest value} \]

Range shows how widely the data values are spread.

Important Formulas and Rules

Mean
\[ \bar{x}=\frac{\sum x}{n} \]

Sum of observations divided by number of observations.

Weighted Mean
\[ \bar{x}=\frac{\sum wx}{\sum w} \]

Used when observations have different weights.

Median Position
\[ \text{Median position}=\frac{n+1}{2} \]

Used when number of observations is odd.

Range
\[ R=H-L \]

\(H\) is highest value and \(L\) is lowest value.

Variance
\[ \sigma^2=\frac{\sum (x-\bar{x})^2}{n} \]

Average of squared deviations from mean.

Standard Deviation
\[ \sigma=\sqrt{\frac{\sum (x-\bar{x})^2}{n}} \]

Square root of variance.

Mean from Frequency
\[ \bar{x}=\frac{\sum fx}{\sum f} \]

Used for frequency distribution.

Mode
\[ \text{Mode}=\text{Most repeated value} \]

Observation with highest frequency.

Even Number Median
\[ \text{Median}=\frac{\left(\frac{n}{2}\right)^{th}+\left(\frac{n}{2}+1\right)^{th}}{2} \]

Average of two middle observations.

Total from Mean
\[ \text{Total}=\text{Mean}\times n \]

Useful when average and number of observations are known.

Deviation
\[ d=x-\bar{x} \]

Difference between observation and mean.

Frequency Total
\[ n=\sum f \]

Total frequency gives total number of observations.

Rule: For median, always arrange data first. For mean, always count the exact number of observations. For mode, check the highest frequency.

Common Types of Questions

Mean-Based Questions

Questions based on average, total value, or missing observation.

  • Find mean
  • Find total
  • Find missing value
  • Weighted average
Median-Based Questions

Questions based on middle value after arranging observations.

  • Odd observations
  • Even observations
  • Arrange data
  • Middle position
Mode-Based Questions

Questions based on the most frequently occurring value.

  • Single mode
  • No mode
  • Multiple modes
  • Frequency table
Spread-Based Questions

Questions based on range, variance, and standard deviation.

  • Range
  • Deviation
  • Variance
  • Standard deviation
Exam approach: For small data sets, arrange values manually. For frequency data, use \(fx\) and total frequency.
Method Bank
Find Mean

Add all observations and divide by count.

\[ \frac{5+10+15}{3}=10 \]
Find Median

Arrange data and take the middle value.

\[ 3,5,7,9,11 \Rightarrow 7 \]
Find Mode

Choose the most repeated value.

\[ 2,4,4,4,6 \Rightarrow 4 \]
Find Range

Subtract lowest value from highest value.

\[ 12-4=8 \]

Tip: For median and range, arranging data in ascending order is the safest first step.

Statistics Solving Flow
Collect Data Arrange Data Apply Formula Interpret Answer Data → arrangement → calculation → interpretation
This flow works for most mean, median, mode, and range problems.
\[ \text{Correct arrangement} \Rightarrow \text{Correct median and range} \]
\[ \text{Correct total} \Rightarrow \text{Correct mean} \]

Step-by-Step Solving Method

Step What to Do Example
Step 1 Read the data carefully. Marks: 10, 20, 30, 40
Step 2 Arrange data if median, mode, or range is required. 4, 7, 9, 12
Step 3 Choose the correct formula. Mean = Sum / Count
Step 4 Substitute values carefully. \( \frac{10+20+30+40}{4} \)
Step 5 Interpret the result according to the question. Average marks = 25
Important: Mean can be affected by very large or very small values, while median is more stable for uneven data.

Solved Examples

Question Method Answer
Find the mean of 10, 20, 30, 40, and 50. Sum of observations:
\[ 10+20+30+40+50=150 \]
Number of observations \(=5\).
\[ \text{Mean}=\frac{150}{5}=30 \]
30
Find the median of 7, 3, 11, 5, and 9. Arrange the data:
\[ 3,5,7,9,11 \]
Middle value:
\[ \text{Median}=7 \]
7
Find the median of 4, 6, 8, 10, 12, and 14. There are 6 observations. Middle two values are 8 and 10.
\[ \text{Median}=\frac{8+10}{2}=9 \]
9
Find the mode of 2, 4, 4, 6, 7, 4, 8. The value 4 appears most frequently.
\[ \text{Mode}=4 \]
4
Find the range of 12, 5, 18, 9, and 20. Highest value \(=20\), lowest value \(=5\).
\[ \text{Range}=20-5=15 \]
15
The mean of 6 numbers is 15. Find their total.
\[ \text{Total}=\text{Mean}\times n \]
\[ \text{Total}=15\times6=90 \]
90
Find the mean of the following frequency table: \(x=2,4,6\), \(f=3,2,5\). Calculate \(fx\):
\[ \sum fx=(2\times3)+(4\times2)+(6\times5)=6+8+30=44 \]
Total frequency:
\[ \sum f=3+2+5=10 \]
Mean:
\[ \bar{x}=\frac{44}{10}=4.4 \]
4.4
Find the variance of 2, 4, and 6. Mean:
\[ \bar{x}=\frac{2+4+6}{3}=4 \]
Squared deviations:
\[ (2-4)^2+(4-4)^2+(6-4)^2=4+0+4=8 \]
Variance:
\[ \sigma^2=\frac{8}{3} \]
\(\frac{8}{3}\)

Note: In statistics questions, always check whether the data is raw data or frequency data.

Common Traps and Shortcuts

Common Traps
  • Finding median without arranging the data.
  • Dividing by wrong number of observations while calculating mean.
  • Confusing mode with median.
  • Using highest value as range instead of subtracting lowest value.
  • Ignoring frequency in frequency distribution.
  • Forgetting to square deviations in variance.
Useful Shortcuts
  • For mean, use total divided by count.
  • For median, arrange data first.
  • For mode, look for the highest frequency.
  • For range, subtract lowest value from highest value.
  • For frequency mean, prepare \(fx\) column.
  • Use total = average × number of observations.
Exam approach: In statistics questions, do not rush. Most errors happen because the data is not arranged or counted properly.

Practice

A) Multiple Choice Questions
  1. Find the mean of 5, 10, and 15.
    5 10 15 20
  2. Find the median of 3, 7, 5, 9, and 11.
    3 5 7 9
  3. Find the mode of 2, 3, 3, 4, 5.
    2 3 4 5
  4. Find the range of 8, 12, 20, and 5.
    10 12 15 20
  5. If the mean of 4 numbers is 12, their total is:
    24 36 48 60
B) Solve the Higher-Order Problems
  1. Find the mean of 12, 18, 24, 30, and 36. (Hint: Add all values and divide by 5.)
  2. Find the median of 14, 8, 20, 6, 18, and 10. (Hint: Arrange first and average the two middle values.)
  3. Find the mode of 5, 7, 7, 9, 10, 7, 12. (Hint: Find the most repeated value.)
  4. The average marks of 8 students is 45. Find the total marks. (Hint: Total = Mean × Number of students.)
  5. Find the mean from \(x=10,20,30\) and \(f=2,3,5\). (Hint: Use \(\bar{x}=\frac{\sum fx}{\sum f}\).)
C) Match the Concept with the Correct Meaning
Concept Correct Meaning
Mean Average of observations
Median Middle value after arranging data
Mode Most repeated value
Range Highest value minus lowest value
Frequency Number of times a value occurs
Variance Average of squared deviations from mean
Aptitude Reminder

Statistics helps us summarize and interpret data. Mean gives the average, median gives the central value, mode gives the most repeated value, and range shows the spread.

Task: Create five questions using mean, median, mode, range, and frequency distribution.

Show Suggested Answers
Multiple Choice
  1. 10
    \[ \text{Mean}=\frac{5+10+15}{3}=10 \]
  2. 7
    Arrange:
    \[ 3,5,7,9,11 \]
    Middle value is \(7\).
  3. 3
    The value \(3\) occurs most frequently.
  4. 15
    Highest value \(=20\), lowest value \(=5\).
    \[ \text{Range}=20-5=15 \]
  5. 48
    \[ \text{Total}=\text{Mean}\times n=12\times4=48 \]
Higher-Order Problems
  1. Mean of 12, 18, 24, 30, and 36:
    \[ \frac{12+18+24+30+36}{5}=\frac{120}{5}=24 \]
    Answer = 24.
  2. Arrange data:
    \[ 6,8,10,14,18,20 \]
    Middle two values are 10 and 14.
    \[ \text{Median}=\frac{10+14}{2}=12 \]
    Answer = 12.
  3. In \(5,7,7,9,10,7,12\), the value \(7\) appears most frequently.
    \[ \text{Mode}=7 \]
    Answer = 7.
  4. Average marks \(=45\), number of students \(=8\).
    \[ \text{Total marks}=45\times8=360 \]
    Answer = 360.
  5. Given \(x=10,20,30\) and \(f=2,3,5\).
    \[ \sum fx=(10\times2)+(20\times3)+(30\times5)=20+60+150=230 \]
    \[ \sum f=2+3+5=10 \]
    \[ \bar{x}=\frac{230}{10}=23 \]
    Answer = 23.
Concept Matching
  1. Mean → Average of observations
  2. Median → Middle value after arranging data
  3. Mode → Most repeated value
  4. Range → Highest value minus lowest value
  5. Frequency → Number of times a value occurs
  6. Variance → Average of squared deviations from mean
Clue Explanation

Mean uses all observations, median depends on position, mode depends on frequency, and range depends only on the highest and lowest values.

Exam tips
  • Arrange data before finding median.
  • Count observations carefully for mean.
  • Mode is based on highest frequency.
  • Range is highest minus lowest.
  • Use \(fx\) table for frequency mean.
  • Do not confuse mean, median, and mode.
Practice MCQs