**Central Limit Theorem: Putting Probability and Statistics Together**^{1, 2, 3}

*Probability is about the likelihood of an event*

Use the theorem to analyze numerical data collected in elicitation conversations with 30 customers. As a result, you can forecast, with 95% certainty, the rate at which the average customer in the target market will adopt my sponsor’s new product. A researcher using the theorem can “draw sweeping and powerful conclusions from relatively little data.” ^{2}

**Two Essential Conditions For Making Confident Forecasts**

*Random Sampling*

A key feature of a random sample of prospective customers is that it does not differ systematically from all potential customers in the target market. During elicitation conversations with a randomly selected prospect, you collect numerical data that measures interest in the product concept. Inferential statisticians refer to such data as scores.

**Sample Size (****n****)= 30 or more**

Sample size is the total number of scores included in the sample. A larger **n** increases the confidence in the forecast of rate of acceptance of the new product by the average customer in the target market

**Scoring a prospect’s interest in the new product**

During elicitation conversations, you gain a measure of the prospect’s interest in the product concept. After the dialogue, score the likelihood their firm will adopt the new product. Scores range from 5.0 to 1.0 depending on the degree interest in the concept.

Interest of prospects in a product concept depends on their ability and motivation to adopt the new product. Everett Rogers’ diffusion of innovation theory helps explain the rate at which a firm will adopt the new product^{ 4}

__Scores__ __Ability and motivation to adopt the new product__

5.0 — **Early Adopter:** Will collaborate in rapid commercial development of the concept. Adopting the new product solves a critical business need and creates exceptional value.

4.0 — **Early Majority:** Will adopt the new product after early adopters show that using the new product solves a major business need and creates good value.

3.0 — **Innovators:** Believe the concept has potential to add value to a business opportunity they are pursuing. Often these opportunities do not align with the capabilities and strategic intent of their firm.

2.0 — **Late Majority:** Will adopt a new product after the bulk to the target market has adopted it. Skeptical about a product concept’s ability to create value for them.

1.0 — **Laggards:** See no value in a product concept’s features

**Analyzing the scores*** ^{3}*

Over the years, statisticians used complicated math to explain why the central limit theorem works. Fortunately, customer researchers don’t need to work with the math. Instead they use straight-forward arithmetic and proven formulas to predict the rate of adoption of a new product by the target market.

*Definitions of the terms In the formulas:*

**n** — Number of elicitation conversations with knowledgeable individuals in the target market

**x** — Score of the ability and motivation of the individual and their firm to adopt the new product

**x _{n}** — Score of the nth elicitation conversation

**x̅** — Sample average (Basic measure of the central tendency in a sample’s distribution

**s** —Standard deviation helps get a sense of the average distance from the sample average

*Calculation formulas *

*Sample average*

**x̅** = **x _{1 +} x_{2+} x_{3 … }x_{n }/_{ }**

**n**

*Standard deviation*

**s** = square root of **[** **(****x _{1}** –

**x̅**

**)**

^{2}**+**

**(**

**x**–

_{2}**x̅**

**)**

^{2 }+ …**(**

**x**–

_{n}**x̅**

**)**

^{2 }/**n**

**–**

**1**

**]**

When you calculate the *sample average* and the *standard deviation* you know, with ~95% certainty, the average rate of adoption of the new product by the target market.

**Endnotes**

- Spieglhalter, D, (2019)
*The Art of Statistics: How to Learn From Data*Hachette Book Group, (New York, NY - Wheelan, C. (2013)
*Naked Statistics: Stripping the Dread From the Data*W. Norton & Company, New York, NY - Frey, B, Ed, (2008)
*Statistics Hacks*O’Reilly Media, Sebastopol, CA

Quote from Frey: “Fortunately, an entire set of scientific tools, the various applications of statistics, can be used to solve the problems caused by our fate-influenced system. Inferential statistics, a field of science based entirely on the nature of probability, allows us to understand the way things work, discover relationships among variables, describe a huge population by seeing just a small bit of it, make uncannily accurate predictions …” - Accessed 09-30-2019,
*Diffusion of innovations*,*Wikipedia*CL