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Statistics

Mean of Grouped Data

What are mean, median, and mode for grouped data? When data is organized into frequency distributions (class intervals with frequencies), we use special formulas to find the three measures of central tendency.

1. Mean (Arithmetic Average) for grouped data: There are three methods to find the mean:

Direct method: Mean = (Σ fᵢxᵢ) / (Σ fᵢ), where xᵢ is the class mark (midpoint) = (upper limit + lower limit)/2
Assumed mean method: Mean = a + (Σ fᵢdᵢ) / (Σ fᵢ), where dᵢ = xᵢ − a (a = assumed mean)
Step deviation method: Mean = a + (Σ fᵢuᵢ) / (Σ fᵢ) × h, where uᵢ = (xᵢ − a)/h, h = class width

2. Median for grouped data: Median is the middle value. For grouped data:

Median = L + [ (N/2 − cf) / f ] × h

where:

L = lower limit of median class
N = total frequency (Σ fᵢ)
cf = cumulative frequency of the class before median class
f = frequency of median class
h = class width

3. Mode for grouped data (bimodal excluded): Mode is the most frequent value. For grouped data:

Mode = L + [ (\(f_{1} - f_{0}\)) / (\(2f_{1} - f_{0} - f_{2}\)) ] × h

where:

L = lower limit of modal class (highest frequency class)
\(f_{1}\) = frequency of modal class
\(f_{0}\) = frequency of class before modal class
\(f_{2}\) = frequency of class after modal class
h = class width

Empirical relationship: For moderately skewed data: Mode ≈ 3 Median − 2 Mean

┌─────────────────────────────────────────────────────────────┐
│      MEAN, MEDIAN, MODE FOR GROUPED DATA - FORMULA MAP       │
└─────────────────────────────────────────────────────────────┘

SAMPLE DATA TABLE:

┌──────────────┬─────────┬──────────┬────────────┬──────────┐
│ Class Interval│ Midpoint│Frequency│  fᵢ × xᵢ   │Cumulative│
│               │   (xᵢ)  │   (fᵢ)  │            │ Frequency│
├──────────────┼─────────┼──────────┼────────────┼──────────┤
│    0-10      │    5    │    2     │     10     │    2     │
│   10-20      │   15    │    5     │     75     │    7     │
│   20-30      │   25    │    8     │    200     │   15     │
│   30-40      │   35    │    4     │    140     │   19     │
│   40-50      │   45    │    1     │     45     │   20     │
├──────────────┼─────────┼──────────┼────────────┼──────────┤
│    Total     │         │   N=20   │   Σfᵢxᵢ=470 │         │
└──────────────┴─────────┴──────────┴────────────┴──────────┘

MEAN = 470/20 = 23.5


FINDING MEDIAN CLASS (N/2 = 10):

    cf just greater than 10 is 15 → median class = 20-30
    L = 20, cf = 7, f = 8, h = 10
    
    Median = 20 + [(10-7)/8] × 10 = 20 + (3/8)×10 = 20 + 3.75 = 23.75


FINDING MODAL CLASS (highest frequency = 8 → 20-30):

    L = 20, f₁=8, f₀=5, f₂=4, h=10
    
    Mode = 20 + [(8-5)/(16-5-4)] × 10 = 20 + [3/7]×10 = 20 + 4.29 = 24.29


COMPARISON ON NUMBER LINE:

    <--- Mean=23.5 --- Median=23.75 --- Mode=24.29 --->
    
    For this positively skewed distribution: Mean < Median < Mode


DIRECT METHOD FLOWCHART:

    Class Intervals → Find Midpoints (xᵢ)
              │
              ▼
    Multiply each xᵢ by frequency fᵢ → fᵢxᵢ
              │
              ▼
    Sum all fᵢxᵢ and sum all fᵢ
              │
              ▼
    Mean = Σfᵢxᵢ / Σfᵢ

Example 1: Find the mean for the following data using the direct method: | Class | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 | |-------|------|-------|-------|-------|-------| | Frequency | 3 | 5 | 7 | 4 | 1 |

Step 1: Find midpoints (xᵢ): 5, 15, 25, 35, 45
Step 2: Calculate fᵢxᵢ: 3×5=15, 5×15=75, 7×25=175, 4×35=140, 1×45=45
Step 3: Σfᵢ = 3+5+7+4+1 = 20
Step 4: Σfᵢxᵢ = 15+75+175+140+45 = 450
Step 5: Mean = 450/20 = 22.5

Answer: Mean = 22.5

Example 2: Find the median for the data in Example 1.

Step 1: N = 20, N/2 = 10
Step 2: Cumulative frequencies: 3, 8, 15, 19, 20
Step 3: Median class = class where cf ≥ 10 → 20-30
Step 4: L = 20, cf = 8, f = 7, h = 10
Step 5: Median = 20 + [(10-8)/7] × 10 = 20 + (2/7)×10 = 20 + 2.86 = 22.86

Answer: Median = 22.86

Example 3: Find the mode for the data in Example 1.

Step 1: Highest frequency = 7 → modal class = 20-30
Step 2: L = 20, \(f_{1}\)=7, \(f_{0}\)=5, \(f_{2}\)=4, h=10
Step 3: Mode = 20 + [(7-5)/(2×7 - 5 - 4)] × 10
Step 4: = 20 + [2/(14-9)] × 10 = 20 + (2/5)×10 = 20 + 4 = 24

Answer: Mode = 24

Quick recap

Mean = Σfᵢxᵢ / Σfᵢ (xᵢ = class midpoint)
Median uses cumulative frequency: Median = L + [(N/2 − cf)/f] × h
Mode uses frequencies of modal and adjacent classes: Mode = L + [(\(f_{1}-f_{0}\))/(\(2f_{1}-f_{0}-f_{2}\))] × h
Modal class = class with highest frequency
Median class = class where cumulative frequency first exceeds N/2
Empirical relation: Mode ≈ 3 Median − 2 Mean (not exact but useful for checking)

✓ Quick check

The mean of 15, 17, 19, 21 and 23 is:

Sum 95 ÷ 5 = 19.

The cumulative frequency of the last class equals:

Cumulating all frequencies gives the total frequency at the last class.

Median of Grouped Data

What is a cumulative frequency table? A cumulative frequency is the running total of frequencies. It tells us "how many observations are less than or equal to a certain value." There are two types:

Less than cumulative frequency: Add frequencies from the lowest class upward
Greater than cumulative frequency: Add frequencies from the highest class downward

How to create a less than cumulative frequency table: 1. List upper limits of each class 2. Add frequencies of all classes with upper limit ≤ the given value

How to create a greater than cumulative frequency table: 1. List lower limits of each class 2. Add frequencies of all classes with lower limit ≥ the given value

Why is this important? Cumulative frequencies are used to:

Draw the ogive (cumulative frequency curve)
Find the median graphically (where the less than and greater than ogives intersect)
Answer questions like "how many students scored less than 60 marks?"

┌─────────────────────────────────────────────────────────────┐
│         CUMULATIVE FREQUENCY TABLE - CONSTRUCTION            │
└─────────────────────────────────────────────────────────────┘

ORIGINAL FREQUENCY TABLE:

┌──────────────┬─────────┬─────────────────────────────────┐
│ Class        │ Frequency│ Calculation                     │
├──────────────┼─────────┼─────────────────────────────────┤
│   0-10       │    5    │                                 │
│  10-20       │    8    │                                 │
│  20-30       │   12    │                                 │
│  30-40       │    7    │                                 │
│  40-50       │    3    │                                 │
└──────────────┴─────────┴─────────────────────────────────┘

LESS THAN CUMULATIVE FREQUENCY TABLE:

┌─────────────────┬─────────┬──────────────────────────────┐
│ Upper Limit     │ Frequency│ Less than Cumulative Frequency│
├─────────────────┼─────────┼──────────────────────────────┤
│     10          │    5    │              5               │
│     20          │    8    │          5 + 8 = 13          │
│     30          │   12    │         13 + 12 = 25         │
│     40          │    7    │         25 + 7 = 32          │
│     50          │    3    │         32 + 3 = 35          │
└─────────────────┴─────────┴──────────────────────────────┘

GREATER THAN CUMULATIVE FREQUENCY TABLE:

┌─────────────────┬─────────┬──────────────────────────────┐
│ Lower Limit     │ Frequency│ Greater than Cumulative Freq │
├─────────────────┼─────────┼──────────────────────────────┤
│      0          │    5    │     35 (all students)        │
│     10          │    8    │     35 - 5 = 30              │
│     20          │   12    │     30 - 8 = 22              │
│     30          │    7    │     22 - 12 = 10             │
│     40          │    3    │     10 - 7 = 3               │
│     50          │    0    │      3 - 3 = 0               │
└─────────────────┴─────────┴──────────────────────────────┘


CUMULATIVE FREQUENCY BAR CHART:

    Cumulative
    Frequency
        35 ┤                                    ■
        30 ┤                              ■─────┘
        25 ┤                        ■─────┘
        20 ┤                  ■─────┘
        15 ┤            ■─────┘
        10 ┤      ■─────┘
         5 ┤■─────┘
         0 └─────┴─────┴─────┴─────┴─────► Upper Limit
            10    20    30    40    50

Example 1: Create a less than cumulative frequency table for the data: | Marks | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 | |-------|------|-------|-------|-------|-------| | Students | 4 | 7 | 10 | 6 | 3 |

Step 1: Upper limits: 10, 20, 30, 40, 50
Step 2: Cumulative frequencies:
Less than 10: 4
Less than 20: 4 + 7 = 11
Less than 30: 11 + 10 = 21
Less than 40: 21 + 6 = 27
Less than 50: 27 + 3 = 30

Answer: (10,4), (20,11), (30,21), (40,27), (50,30)

Example 2: From the cumulative frequency table in Example 1, find how many students scored less than 35 marks.

Step 1: 35 lies in class 30-40
Step 2: Students with marks < 30 = 21
Step 3: In class 30-40, marks 30-35 is half the class (assuming uniform distribution)
Step 4: Students in 30-40 = 6, half = 3
Step 5: Total less than 35 ≈ 21 + 3 = 24

Answer: Approximately 24 students

Example 3: Create a greater than cumulative frequency table for Example 1.

Step 1: Lower limits: 0, 10, 20, 30, 40, 50
Step 2: Greater than cumulative:
Greater than 0: 30 (all)
Greater than 10: 30 − 4 = 26
Greater than 20: 26 − 7 = 19
Greater than 30: 19 − 10 = 9
Greater than 40: 9 − 6 = 3
Greater than 50: 3 − 3 = 0

Answer: (0,30), (10,26), (20,19), (30,9), (40,3), (50,0)

Quick recap

Less than cumulative frequency: Sum of frequencies up to the upper limit of each class
Greater than cumulative frequency: Sum of frequencies from the lower limit to the end
Less than cf is increasing, greater than cf is decreasing
The last less than cf = total frequency = first greater than cf
Use less than cf to find "how many scored less than x"
Cumulative frequency tables help draw ogives (smooth curves)

✓ Quick check

The mean of 4, 6, 8, 10 and 12 is:

(4 + 6 + 8 + 10 + 12) ÷ 5 = 40 ÷ 5 = 8.

The class size of the interval 15.5–20.5 is:

Class size = 20.5 − 15.5 = 5.

Mode and Ogives

What are histograms, frequency polygons, and ogives? These are graphical representations of frequency distributions that help visualize data patterns.

1. Histogram: A histogram is a bar graph where:

Bars are drawn without gaps (touching each other)
Width of each bar = class width
Height of each bar = frequency of that class
The x-axis shows class boundaries, y-axis shows frequency

2. Frequency Polygon: A frequency polygon is a line graph created by:

Plotting points at class midpoints with heights equal to frequencies
Joining these points with straight lines
Adding two extra points at the ends (with zero frequency) to close the polygon

3. Ogive (Cumulative Frequency Curve): An ogive is a smooth curve drawn using cumulative frequencies:

Less than ogive: Plot upper limits vs less than cumulative frequency, then join with a smooth curve
Greater than ogive: Plot lower limits vs greater than cumulative frequency, then join with a smooth curve

Finding median from ogives: The x-coordinate of the intersection of the less than and greater than ogives gives the median.

┌─────────────────────────────────────────────────────────────┐
│       HISTOGRAM, FREQUENCY POLYGON, AND OGIVE - PLOTS        │
└─────────────────────────────────────────────────────────────┘

SAMPLE DATA: Class: 0-10,10-20,20-30,30-40,40-50; Freq: 5,8,12,7,3

HISTOGRAM (Bars touching):

    Frequency
       12 ┤        ┌─────┐
       10 ┤        │     │
        8 ┤   ┌────┤     │
        6 ┤   │    │     │
        4 ┤   │    │     │   ┌─┐
        2 ┤   │    │     │   │ │
        0 └───┴────┴─────┴───┴─┴──► Class
            0-10 10-20 20-30 30-40 40-50


FREQUENCY POLYGON (Points at midpoints: 5,15,25,35,45)

    Frequency
       12 ┤        ●
       10 ┤       / \
        8 ┤      /   \
        6 ┤     /     \
        4 ┤    /       \
        2 ┤   /         \
        0 ┼──●───────────●──●──►
           -5  5  15 25 35 45 55
              (Extra points at 0 and 50 to close polygon)


OGIVE (Less than cumulative curve):

    Cumulative
    Frequency
       35 ┤                              ●
       30 ┤                        ●─────┘
       25 ┤                  ●─────┘
       20 ┤            ●─────┘
       15 ┤      ●─────┘
       10 ┤●─────┘
        5 ┤
        0 └─────┴─────┴─────┴─────┴─────► Upper Limit
           10    20    30    40    50


OGIVE (Both less than and greater than on same axes):

    Cumulative
    Frequency
       35 ┤  ●                              
       30 ┤  │  ○                          
       25 ┤  │  │  ●                      
       20 ┤  │  │  │  ○                  
       15 ┤  │  │  │  │  ●              
       10 ┤  │  │  │  │  │  ○          
        5 ┤  │  │  │  │  │  │  ●      
        0 └──┼──┼──┼──┼──┼──┼──►
            10 20 30 40 50 60
           Less than (●)  Greater than (○)
           
           INTERSECTION point gives MEDIAN!

Example 1: Draw a histogram for the data: Class 0-10 (f=4), 10-20 (f=6), 20-30 (f=10), 30-40 (f=5).

Step 1: Identify class boundaries: 0-10, 10-20, 20-30, 30-40
Step 2: Bars have width 10, heights = frequencies
Step 3: Bars touch each other (no gaps)

Answer: Histogram drawn with bars at heights 4,6,10,5

Example 2: From the following less than ogive points, estimate the median: (10,4), (20,10), (30,22), (40,30), (50,35)

Step 1: Total frequency N = 35
Step 2: N/2 = 17.5
Step 3: On ogive, find x-coordinate where cumulative frequency = 17.5
Step 4: 17.5 lies between (20,10) and (30,22)
Step 5: Interpolate: 20 + [(17.5-10)/(22-10)] × 10 = 20 + (7.5/12)×10 = 20 + 6.25 = 26.25

Answer: Median ≈ 26.25

Example 3: Using the data in Example 1, find the number of students scoring between 15 and 35 marks (using frequency polygon).

Step 1: Midpoints: 5, 15, 25, 35; frequencies: 4, 6, 10, 5
Step 2: Frequencies for classes containing 15-35: class 10-20 (15 is midpoint), 20-30, 30-40 (35 is midpoint)
Step 3: These classes cover 15 to 35 completely
Step 4: Total frequency = 6 + 10 + 5 = 21 students

Answer: 21 students (within 15-35 marks)

Quick recap

Histogram: Touching bars for continuous data; area of bar represents frequency
Frequency polygon: Points at midpoints joined by lines; starts and ends at zero frequency
Less than ogive: Upper limits vs less than cumulative frequency
Greater than ogive: Lower limits vs greater than cumulative frequency
Median can be found at intersection of the two ogives
Ogives are useful for finding percentiles and quartiles
Histograms show distribution shape (symmetric, skewed left, skewed right)

✓ Quick check

A student scores 85, 90, 78, 92 and 80. The mean score is:

(85 + 90 + 78 + 92 + 80) ÷ 5 = 425 ÷ 5 = 85.

Temperatures (°C) over 5 days are 28, 30, 32, 31 and 29. The mean temperature is:

(28 + 30 + 32 + 31 + 29) ÷ 5 = 150 ÷ 5 = 30 °C.

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